diff --git "a/minictx-valid/pnt.jsonl" "b/minictx-valid/pnt.jsonl"
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+++ "b/minictx-valid/pnt.jsonl"
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+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\nlemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]\n\nlemma sum_eq_int_deriv_aux_lt {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_lt_kpOne : b < k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k := Int.floor_eq_iff.mpr \u27e8by linarith [ha.1, ha.2], by linarith\u27e9\n  simp only [flb_eq_k, gt_iff_lt, lt_add_iff_pos_right, zero_lt_one, Finset.Icc_eq_empty_of_lt,\n    Finset.sum_empty]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont]\n  have : Finset.Ioc k k = {} := by simp only [ge_iff_le, le_refl, Finset.Ioc_eq_empty_of_le]\n  simp only [this, Finset.sum_empty, one_div]; ring_nf\n\nlemma sum_eq_int_deriv_aux1 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  by_cases h : b = k + 1\n  \u00b7 exact sum_eq_int_deriv_aux_eq h \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact sum_eq_int_deriv_aux_lt ha (Ne.lt_of_le h b_le_kpOne) \u03c6Diff deriv\u03c6Cont\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv_aux]\\label{sum_eq_int_deriv_aux}\\lean{sum_eq_int_deriv_aux}\\leanok\n  Let $k \\le a < b\\le k+1$, with $k$ an integer, and let $\\phi$ be continuously differentiable on\n  $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma sum_eq_int_deriv_aux {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  have fl_a_eq_k : \u230aa\u230b = k := Int.floor_eq_iff.mpr \u27e8ha.1, by linarith [ha.2]\u27e9\n  convert sum_eq_int_deriv_aux1 ha b_le_kpOne \u03c6Diff deriv\u03c6Cont using 2\n  \u00b7 rw [fl_a_eq_k]\n  \u00b7 congr\n  \u00b7 apply intervalIntegral.integral_congr_ae\n    have : \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.volume, x \u2260 b := by\n      convert Countable.ae_not_mem (s := {b}) (by simp) (\u03bc := MeasureTheory.volume) using 1\n    filter_upwards [this]\n    intro x x_ne_b hx\n    rw [uIoc_of_le ha.2.le, mem_Ioc] at hx\n    congr\n    exact Int.floor_eq_iff.mpr \u27e8by linarith [ha.1], by have := Ne.lt_of_le x_ne_b hx.2; linarith\u27e9\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\n-- Thanks to Arend Mellendijk\n\nlemma interval_induction_aux_int (n : \u2115) : \u2200 (P : \u211d \u2192 \u211d \u2192 Prop)\n    (_ : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (_ : \u2200 (a : \u211d) (k : \u2124) (c : \u211d), a < k \u2192 k < c \u2192 P a k \u2192 P k c \u2192 P a c)\n    (a b : \u211d) (_ : a < b) (_ : n = \u230ab\u230b - \u230aa\u230b),\n    P a b := by\n  induction n using Nat.case_strong_induction_on with\n  | hz =>\n    intro P base _ a b hab hn\n    apply base a b \u230aa\u230b (Int.floor_le a) hab\n    rw [(by simp only [CharP.cast_eq_zero] at hn; linarith : \u230aa\u230b = \u230ab\u230b)]\n    exact (Int.lt_floor_add_one b).le\n  | hi n ih =>\n    intro P base step a b _ hn\n    have Pa : P a (\u230aa\u230b + 1) :=\n      base a (\u230aa\u230b + 1) \u230aa\u230b (Int.floor_le a) (Int.lt_floor_add_one a) (le_of_eq rfl)\n    by_cases b_le_flaP1 : b = \u230aa\u230b + 1\n    \u00b7 rwa [b_le_flaP1]\n    have flaP1_lt_b : \u230aa\u230b + 1 < b := by\n      simp only [Nat.cast_succ] at hn\n      have : (\u230aa\u230b : \u211d) + 1 \u2264 \u230ab\u230b := by exact_mod_cast (by linarith)\n      exact Ne.lt_of_le (id (Ne.symm b_le_flaP1)) (by linarith [Int.floor_le b] : \u230aa\u230b + 1 \u2264 b)\n    have Pfla_b : P (\u230aa\u230b + 1) b := by\n      apply ih n (le_of_eq rfl) P base step (\u230aa\u230b + 1) b flaP1_lt_b\n      simp only [Int.floor_add_one, Int.floor_intCast, Nat.cast_succ] at hn \u22a2\n      linarith\n    refine step a (\u230aa\u230b + 1) b ?_ (by exact_mod_cast flaP1_lt_b) (by exact_mod_cast Pa)\n      (by exact_mod_cast Pfla_b)\n    have := Int.lt_floor_add_one a\n    exact_mod_cast this\n\nlemma interval_induction (P : \u211d \u2192 \u211d \u2192 Prop)\n    (base : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (step : \u2200 (a : \u211d) (k : \u2124) (b : \u211d), a < k \u2192 k < b \u2192 P a k \u2192 P k b \u2192 P a b)\n    (a b : \u211d) (hab : a < b) : P a b := by\n  set n := \u230ab\u230b - \u230aa\u230b with hn\n  clear_value n\n  have : 0 \u2264 n := by simp only [hn, sub_nonneg, ge_iff_le, Int.floor_le_floor _ _ (hab.le)]\n  lift n to \u2115 using this\n  exact interval_induction_aux_int n P base step a b hab hn\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv]\\label{sum_eq_int_deriv}\\lean{sum_eq_int_deriv}\\leanok\n  Let $a < b$, and let $\\phi$ be continuously differentiable on $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\n/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {\u03b1 : Type*} [LinearOrder \u03b1] [LocallyFiniteOrder \u03b1]\n    {a b c: \u03b1} [DecidableEq \u03b1] (hb : b \u2208 Icc a c) : Ioc a b = Ioc a c \\ Ioc b c := by\n  ext x\n  simp only [mem_Ioc, mem_sdiff, not_and, not_le]\n  constructor\n  \u00b7 refine fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8\u27e8h\u2081, le_trans h\u2082 (mem_Icc.mp hb).2\u27e9, by contrapose! h\u2082; exact h\u2082.1\u27e9\n  \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8h\u2081.1, by contrapose! h\u2082; exact \u27e8h\u2082, h\u2081.2\u27e9\u27e9\n\n-- In Ya\u00ebl Dillies's API (https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Esum_add_adjacent_intervals/near/430127101)\nlemma Finset.sum_Ioc_add_sum_Ioc {a b c : \u2124} (f : \u2124 \u2192 \u2102) (hb : b \u2208 Icc a c):\n    (\u2211 n in Finset.Ioc a b, f n) + (\u2211 n in Finset.Ioc b c, f n) = \u2211 n in Finset.Ioc a c, f n := by\n  convert Finset.sum_sdiff (s\u2081 := Finset.Ioc b c) (s\u2082 := Finset.Ioc a c) ?_\n  \u00b7 exact Finset.Ioc_diff_Ioc hb\n  \u00b7 exact Finset.Ioc_subset_Ioc (mem_Icc.mp hb).1 (by rfl)\n\nlemma integrability_aux\u2080 {a b : \u211d} :\n    \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.Measure.restrict MeasureTheory.volume [[a, b]],\n      \u2016(\u230ax\u230b : \u2102)\u2016 \u2264 max \u2016a\u2016 \u2016b\u2016 + 1 := by\n  apply (MeasureTheory.ae_restrict_iff' measurableSet_Icc).mpr\n  refine MeasureTheory.ae_of_all _ (fun x hx \u21a6 ?_)\n  simp only [inf_le_iff, le_sup_iff, mem_Icc] at hx\n  simp only [norm_int, Real.norm_eq_abs]\n  have : |x| \u2264 max |a| |b| := by\n    cases' hx.1 with x_ge_a x_ge_b <;> cases' hx.2 with x_le_a x_le_b\n    \u00b7 rw [(by linarith : x = a)]; apply le_max_left\n    \u00b7 apply abs_le_max_abs_abs x_ge_a x_le_b\n    \u00b7 rw [max_comm]; apply abs_le_max_abs_abs x_ge_b x_le_a\n    \u00b7 rw [(by linarith : x = b)]; apply le_max_right\n  cases' abs_cases x with hx hx\n  \u00b7 rw [_root_.abs_of_nonneg <| by exact_mod_cast Int.floor_nonneg.mpr hx.2]\n    apply le_trans (Int.floor_le x) <| le_trans (hx.1 \u25b8 this) (by simp)\n  \u00b7 rw [_root_.abs_of_nonpos <| by exact_mod_cast Int.floor_nonpos hx.2.le]\n    linarith [(Int.lt_floor_add_one x).le]\n\nlemma integrability_aux\u2081 {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (\u230ax\u230b : \u2102)) MeasureTheory.volume a b := by\n  rw [intervalIntegrable_iff']\n  apply MeasureTheory.Measure.integrableOn_of_bounded ?_ ?_ integrability_aux\u2080\n  \u00b7 simp only [Real.volume_interval, ne_eq, ENNReal.ofReal_ne_top, not_false_eq_true]\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply Measurable.comp (by exact fun \u2983t\u2984 _ \u21a6 trivial) Int.measurable_floor\n\nlemma integrability_aux\u2082 {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (1 : \u2102) / 2 - x) MeasureTheory.volume a b :=\n  ContinuousOn.intervalIntegrable <| Continuous.continuousOn (by continuity)\n\nlemma integrability_aux {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (\u230ax\u230b : \u2102) + 1 / 2 - x) MeasureTheory.volume a b := by\n  convert integrability_aux\u2081.add integrability_aux\u2082 using 2; ring\n\nlemma uIcc_subsets {a b c : \u211d} (hc : c \u2208 Icc a b) :\n    [[a, c]] \u2286 [[a, b]] \u2227 [[c, b]] \u2286 [[a, b]] := by\n  constructor <;> rw [uIcc_of_le ?_, uIcc_of_le ?_]\n  any_goals apply Icc_subset_Icc\n  all_goals linarith [hc.1, hc.2]\n\nlemma sum_eq_int_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (a_lt_b : a < b)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n      (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n        - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  let P := fun a\u2081 b\u2081 \u21a6 (\u2200 x \u2208 [[a\u2081, b\u2081]], HasDerivAt \u03c6 (deriv \u03c6 x) x) \u2192\n    (ContinuousOn (deriv \u03c6) [[a\u2081, b\u2081]]) \u2192\n    \u2211 n in Finset.Ioc \u230aa\u2081\u230b \u230ab\u2081\u230b, \u03c6 n =\n    (\u222b x in a\u2081..b\u2081, \u03c6 x) + (\u230ab\u2081\u230b + 1 / 2 - b\u2081) * \u03c6 b\u2081 - (\u230aa\u2081\u230b + 1 / 2 - a\u2081) * \u03c6 a\u2081\n      - \u222b x in a\u2081..b\u2081, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x\n  apply interval_induction P ?base ?step a b a_lt_b \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact fun _ _ _ k\u2081_le_a\u2081 a\u2081_lt_b\u2081 b\u2081_le_k\u2081 \u03c6Diff\u2081 deriv\u03c6Cont\u2081 \u21a6\n      sum_eq_int_deriv_aux \u27e8k\u2081_le_a\u2081, a\u2081_lt_b\u2081\u27e9 b\u2081_le_k\u2081 \u03c6Diff\u2081 deriv\u03c6Cont\u2081\n  \u00b7 intro a\u2081 k\u2081 b\u2081 a\u2081_lt_k\u2081 k\u2081_lt_b\u2081 ih\u2081 ih\u2082 \u03c6Diff\u2081 deriv\u03c6Cont\u2081\n    have subs := uIcc_subsets \u27e8a\u2081_lt_k\u2081.le, k\u2081_lt_b\u2081.le\u27e9\n    have s\u2081 := ih\u2081 (fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.1 hx) <| deriv\u03c6Cont\u2081.mono subs.1\n    have s\u2082 := ih\u2082 (fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.2 hx) <| deriv\u03c6Cont\u2081.mono subs.2\n    convert Mathlib.Tactic.LinearCombination.add_pf s\u2081 s\u2082 using 1\n    \u00b7 rw [\u2190 Finset.sum_Ioc_add_sum_Ioc]\n      simp only [Finset.mem_Icc, Int.floor_intCast, Int.le_floor]\n      exact \u27e8Int.cast_le.mp <| le_trans (Int.floor_le a\u2081) a\u2081_lt_k\u2081.le, k\u2081_lt_b\u2081.le\u27e9\n    \u00b7 set I\u2081 := \u222b (x : \u211d) in a\u2081..b\u2081, \u03c6 x\n      set I\u2082 := \u222b (x : \u211d) in a\u2081..k\u2081, \u03c6 x\n      set I\u2083 := \u222b (x : \u211d) in k\u2081..b\u2081, \u03c6 x\n      set J\u2081 := \u222b (x : \u211d) in a\u2081..b\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      set J\u2082 := \u222b (x : \u211d) in a\u2081..k\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      set J\u2083 := \u222b (x : \u211d) in k\u2081..b\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      have hI : I\u2082 + I\u2083 = I\u2081 := by\n        apply intervalIntegral.integral_add_adjacent_intervals <;>\n        apply ContinuousOn.intervalIntegrable\n        \u00b7 exact HasDerivAt.continuousOn <| fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.1 hx\n        \u00b7 exact HasDerivAt.continuousOn <| fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.2 hx\n      have hJ : J\u2082 + J\u2083 = J\u2081 := by\n        apply intervalIntegral.integral_add_adjacent_intervals <;>\n        apply IntervalIntegrable.mul_continuousOn\n        any_goals apply integrability_aux\n        \u00b7 exact deriv\u03c6Cont\u2081.mono subs.1\n        \u00b7 exact deriv\u03c6Cont\u2081.mono subs.2\n      rw [\u2190 hI, \u2190 hJ]; ring\n/-%%\n\\begin{proof}\\uses{sum_eq_int_deriv_aux}\\leanok\n  Apply Lemma \\ref{sum_eq_int_deriv_aux} in blocks of length $\\le 1$.\n\\end{proof}\n%%-/\n\nlemma xpos_of_uIcc {a b : \u2115} (ha : a \u2208 Ioo 0 b) {x : \u211d} (x_in : x \u2208 [[(a : \u211d), b]]) :\n    0 < x := by\n  rw [uIcc_of_le (by exact_mod_cast ha.2.le), mem_Icc] at x_in\n  linarith [(by exact_mod_cast ha.1 : (0 : \u211d) < a)]\n\nlemma neg_s_ne_neg_one {s : \u2102} (s_ne_one : s \u2260 1) : -s \u2260 -1 := fun hs \u21a6 s_ne_one <| neg_inj.mp hs\n\nlemma ZetaSum_aux1\u2081 {a b : \u2115} {s : \u2102} (s_ne_one : s \u2260 1) (ha : a \u2208 Ioo 0 b) :\n    (\u222b (x : \u211d) in a..b, 1 / (x : \u2102) ^ s) =\n    (b ^ (1 - s) - a ^ (1 - s)) / (1 - s) := by\n  convert integral_cpow (a := a) (b := b) (r := -s) ?_ using 1\n  \u00b7 refine intervalIntegral.integral_congr fun x hx \u21a6 one_div_cpow_eq ?_\n    exact (xpos_of_uIcc ha hx).ne'\n  \u00b7 norm_cast; rw [(by ring : -s + 1 = 1 - s)]\n  \u00b7 right; refine \u27e8neg_s_ne_neg_one s_ne_one, ?_\u27e9\n    exact fun hx \u21a6 (lt_self_iff_false 0).mp <| xpos_of_uIcc ha hx\n\nlemma ZetaSum_aux1\u03c6Diff {s : \u2102} {x : \u211d} (xpos : 0 < x) :\n    HasDerivAt (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) (deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) x) x := by\n  apply hasDerivAt_deriv_iff.mpr <| DifferentiableAt.div (differentiableAt_const _) ?_ ?_\n  \u00b7 exact Real.differentiableAt_cpow_const_of_ne s xpos\n  \u00b7 simp [cpow_eq_zero_iff, xpos.ne']\n\nlemma ZetaSum_aux1\u03c6deriv {s : \u2102} (s_ne_zero : s \u2260 0) {x : \u211d} (xpos : 0 < x) :\n    deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) x = (fun (x : \u211d) \u21a6 -s * (x : \u2102) ^ (-(s + 1))) x := by\n  let r := -s - 1\n  have r_add1_ne_zero : r + 1 \u2260 0 := fun hr \u21a6 by simp [neg_ne_zero.mpr s_ne_zero, r] at hr\n  have r_ne_neg1 : r \u2260 -1 := fun hr \u21a6 (hr \u25b8 r_add1_ne_zero) <| by norm_num\n  have hasDeriv := hasDerivAt_ofReal_cpow xpos.ne' r_ne_neg1\n  have := hasDeriv.deriv \u25b8 deriv_const_mul (-s) (hasDeriv).differentiableAt\n  convert this using 2\n  \u00b7 ext y\n    by_cases y_zero : (y : \u2102) = 0\n    \u00b7 simp only [y_zero, ofReal_zero, ne_eq, s_ne_zero, not_false_eq_true, zero_cpow, div_zero,\n      r_add1_ne_zero, zero_div, mul_zero]\n    \u00b7 have : (y : \u2102) ^ s \u2260 0 := fun hy \u21a6 y_zero ((cpow_eq_zero_iff _ _).mp hy).1\n      field_simp [r, mul_assoc, \u2190 Complex.cpow_add]\n  \u00b7 ring_nf\n\nlemma ZetaSum_aux1deriv\u03c6Cont {s : \u2102} (s_ne_zero : s \u2260 0) {a b : \u2115} (ha : a \u2208 Ioo 0 b) :\n    ContinuousOn (deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s)) [[a, b]] := by\n  have : EqOn _ (fun (t : \u211d) \u21a6 -s * (t : \u2102) ^ (-(s + 1))) [[a, b]] :=\n    fun x hx \u21a6 ZetaSum_aux1\u03c6deriv s_ne_zero <| xpos_of_uIcc ha hx\n  refine ContinuousOn.congr ?_ this\n  refine (ContinuousOn.cpow_const continuous_ofReal.continuousOn ?_).const_smul (c := -s)\n  exact fun x hx \u21a6 ofReal_mem_slitPlane.mpr <| xpos_of_uIcc ha hx\n\n/-%%\n\\begin{lemma}[ZetaSum_aux1]\\label{ZetaSum_aux1}\\lean{ZetaSum_aux1}\\leanok\n  Let $0 < a < b$ be natural numbers and $s\\in \\C$ with $s \\ne 1$ and $s \\ne 0$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\frac{1}{n^s} =  \\frac{b^{1-s} - a^{1-s}}{1-s} + \\frac{b^{-s}-a^{-s}}{2} + s \\int_a^b \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma ZetaSum_aux1 {a b : \u2115} {s : \u2102} (s_ne_one : s \u2260 1) (s_ne_zero : s \u2260 0) (ha : a \u2208 Ioo 0 b) :\n    \u2211 n in Finset.Ioc (a : \u2124) b, 1 / (n : \u2102) ^ s =\n    (b ^ (1 - s) - a ^ (1 - s)) / (1 - s) + 1 / 2 * (1 / b ^ (s)) - 1 / 2 * (1 / a ^ s)\n      + s * \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1)) := by\n  let \u03c6 := fun (x : \u211d) \u21a6 1 / (x : \u2102) ^ s\n  let \u03c6' := fun (x : \u211d) \u21a6 -s * (x : \u2102) ^ (-(s + 1))\n  have xpos : \u2200 x \u2208 [[(a : \u211d), b]], 0 < x := fun x hx \u21a6 xpos_of_uIcc ha hx\n  have \u03c6Diff : \u2200 x \u2208 [[(a : \u211d), b]], HasDerivAt \u03c6 (deriv \u03c6 x) x := fun x hx \u21a6 ZetaSum_aux1\u03c6Diff (xpos x hx)\n  have \u03c6deriv : \u2200 x \u2208 [[(a : \u211d), b]], deriv \u03c6 x = \u03c6' x := by\n    exact fun x hx \u21a6 ZetaSum_aux1\u03c6deriv s_ne_zero (xpos x hx)\n  have deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]] := ZetaSum_aux1deriv\u03c6Cont s_ne_zero ha\n  convert sum_eq_int_deriv (by exact_mod_cast ha.2) \u03c6Diff deriv\u03c6Cont using 1\n  \u00b7 congr <;> simp only [Int.floor_natCast]\n  \u00b7 rw [Int.floor_natCast, Int.floor_natCast, \u2190 intervalIntegral.integral_const_mul]\n    simp_rw [mul_div, \u2190 mul_div, ZetaSum_aux1\u2081 s_ne_one ha]\n    conv => rhs; rw [sub_eq_add_neg]\n    congr; any_goals norm_cast; simp only [one_div, add_sub_cancel_left]\n    rw [\u2190 intervalIntegral.integral_neg, intervalIntegral.integral_congr]\n    intro x hx; simp_rw [\u03c6deriv x hx, \u03c6']; ring_nf\n/-%%\n\\begin{proof}\\uses{sum_eq_int_deriv}\\leanok\n  Apply Lemma \\ref{sum_eq_int_deriv} to the function $x \\mapsto x^{-s}$.\n\\end{proof}\n%%-/\n\nlemma ZetaSum_aux1_1' {a b x : \u211d} (apos : 0 < a) (hx : x \u2208 Icc a b)\n    : 0 < x := lt_of_lt_of_le apos hx.1\n\nlemma ZetaSum_aux1_1 {a b x : \u211d} (apos : 0 < a) (a_lt_b : a < b) (hx : x \u2208 [[a,b]])\n    : 0 < x :=  lt_of_lt_of_le apos (uIcc_of_le a_lt_b.le \u25b8 hx).1\n\nlemma ZetaSum_aux1_2 {a b : \u211d} {c : \u211d} (apos : 0 < a) (a_lt_b : a < b)\n    (h : c \u2260 0 \u2227 0 \u2209 [[a, b]]) :\n    \u222b (x : \u211d) in a..b, 1 / x ^ (c+1) = (a ^ (-c) - b ^ (-c)) / c := by\n  rw [(by ring : (a ^ (-c) - b ^ (-c)) / c = (b ^ (-c) - a ^ (-c)) / (-c))]\n  have := integral_rpow (a := a) (b := b) (r := -c-1) (Or.inr \u27e8by simp [h.1], h.2\u27e9)\n  simp only [sub_add_cancel] at this\n  rw [\u2190 this]\n  apply intervalIntegral.integral_congr\n  intro x hx\n  have : 0 \u2264 x := (ZetaSum_aux1_1 apos a_lt_b hx).le\n  simp [div_rpow_eq_rpow_neg _ _ _ this, sub_eq_add_neg, add_comm]\n\nlemma ZetaSum_aux1_3a (x : \u211d) : -(1/2) < \u230a x \u230b + 1/2 - x := by\n  norm_num [\u2190 add_assoc]; linarith [sub_pos_of_lt (Int.lt_floor_add_one x)]\n\nlemma ZetaSum_aux1_3b (x : \u211d) : \u230ax\u230b + 1/2 - x \u2264 1/2 := by\n  ring_nf; exact add_le_of_nonpos_right <| sub_nonpos.mpr (Int.floor_le x)\n\nlemma ZetaSum_aux1_3 (x : \u211d) : |(\u230ax\u230b + 1/2 - x)| \u2264 1/2 :=\n  abs_le.mpr \u27e8le_of_lt (ZetaSum_aux1_3a x), ZetaSum_aux1_3b x\u27e9\n\nlemma ZetaSum_aux1_4' (x : \u211d) (hx : 0 < x) (s : \u2102) :\n      \u2016(\u230ax\u230b + 1 / 2 - (x : \u211d)) / (x : \u2102) ^ (s + 1)\u2016 =\n      |\u230ax\u230b + 1 / 2 - x| / x ^ ((s + 1).re) := by\n  simp [map_div\u2080, abs_ofReal, Complex.abs_cpow_eq_rpow_re_of_pos hx, \u2190 abs_ofReal]\n\nlemma ZetaSum_aux1_4 {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} :\n  \u222b (x : \u211d) in a..b, \u2016(\u2191\u230ax\u230b + (1 : \u211d) / 2 - \u2191x) / (x : \u2102) ^ (s + 1)\u2016 =\n    \u222b (x : \u211d) in a..b, |\u230ax\u230b + 1 / 2 - x| / x ^ (s + 1).re := by\n  apply intervalIntegral.integral_congr\n  exact fun x hx \u21a6 ZetaSum_aux1_4' x (ZetaSum_aux1_1 apos a_lt_b hx) s\n\nlemma ZetaSum_aux1_5a {a b : \u211d} (apos : 0 < a) {s : \u2102} (x : \u211d)\n  (h : x \u2208 Icc a b) : |\u2191\u230ax\u230b + 1 / 2 - x| / x ^ (s.re + 1) \u2264 1 / x ^ (s.re + 1) := by\n  apply div_le_div_of_nonneg_right _ _\n  \u00b7 exact le_trans (ZetaSum_aux1_3 x) (by norm_num)\n  \u00b7 apply Real.rpow_nonneg <| le_of_lt (ZetaSum_aux1_1' apos h)\n\nlemma ZetaSum_aux1_5b {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} (\u03c3pos : 0 < s.re) :\n  IntervalIntegrable (fun u \u21a6 1 / u ^ (s.re + 1)) MeasureTheory.volume a b := by\n  apply ContinuousOn.intervalIntegrable_of_Icc (le_of_lt a_lt_b) _\n  apply ContinuousOn.div continuousOn_const\n  \u00b7 refine ContinuousOn.rpow_const continuousOn_id ?_\n    exact fun x hx \u21a6 Or.inl (ne_of_gt <| ZetaSum_aux1_1' apos hx)\n  \u00b7 exact fun x hx h \u21a6 by rw [Real.rpow_eq_zero] at h <;> linarith [ZetaSum_aux1_1' apos hx]\n\nlemma ZetaSum_aux1_5c {a b : \u211d} {s : \u2102} :\n    let g : \u211d \u2192 \u211d := fun u \u21a6 |\u2191\u230au\u230b + 1 / 2 - u| / u ^ (s.re + 1);\n    MeasureTheory.AEStronglyMeasurable g\n      (MeasureTheory.Measure.restrict MeasureTheory.volume (\u0399 a b)) := by\n  intro\n  refine (Measurable.div ?_ <| measurable_id.pow_const _).aestronglyMeasurable\n  refine (_root_.continuous_abs).measurable.comp ?_\n  refine Measurable.sub (Measurable.add ?_ measurable_const) measurable_id\n  exact Measurable.comp (by exact fun _ _ \u21a6 trivial) Int.measurable_floor\n\nlemma ZetaSum_aux1_5d {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} (\u03c3pos : 0 < s.re) :\n  IntervalIntegrable (fun u \u21a6 |\u2191\u230au\u230b + 1 / 2 - u| / u ^ (s.re + 1)) MeasureTheory.volume a b := by\n  set g : \u211d \u2192 \u211d := (fun u \u21a6 |\u2191\u230au\u230b + 1 / 2 - u| / u ^ (s.re + 1))\n  apply IntervalIntegrable.mono_fun (ZetaSum_aux1_5b apos a_lt_b \u03c3pos) ZetaSum_aux1_5c ?_\n  filter_upwards with x\n  simp only [g, Real.norm_eq_abs, one_div, norm_inv, abs_div, _root_.abs_abs]\n  conv => rw [div_eq_mul_inv, \u2190 one_div]; rhs; rw [\u2190 one_mul |x ^ (s.re + 1)|\u207b\u00b9]\n  refine mul_le_mul ?_ (le_refl _) (by simp) <| by norm_num\n  exact le_trans (ZetaSum_aux1_3 x) <| by norm_num\n\nlemma ZetaSum_aux1_5 {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} (\u03c3pos : 0 < s.re) :\n  \u222b (x : \u211d) in a..b, |\u230ax\u230b + 1 / 2 - x| / x ^ (s.re + 1) \u2264 \u222b (x : \u211d) in a..b, 1 / x ^ (s.re + 1) := by\n  apply intervalIntegral.integral_mono_on (le_of_lt a_lt_b) ?_ ?_\n  \u00b7 exact ZetaSum_aux1_5a apos\n  \u00b7 exact ZetaSum_aux1_5d apos a_lt_b \u03c3pos\n  \u00b7 exact ZetaSum_aux1_5b apos a_lt_b \u03c3pos\n\n/-%%\n\\begin{lemma}[ZetaSum_aux1a]\\label{ZetaSum_aux1a}\\lean{ZetaSum_aux1a}\\leanok\nFor any $0 < a < b$ and  $s \\in \\C$ with $\\sigma=\\Re(s)>0$,\n$$\n\\left|\\int_a^b \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\\right|\n\\le \\frac{a^{-\\sigma}-b^{-\\sigma}}{\\sigma}.\n$$\n\\end{lemma}\n%%-/\nlemma ZetaSum_aux1a {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} (\u03c3pos : 0 < s.re) :\n    \u2016\u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\u2016 \u2264\n      (a ^ (-s.re) - b ^ (-s.re)) / s.re := by\n  calc\n    _ \u2264 \u222b x in a..b, \u2016(\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\u2016 := ?_\n    _ = \u222b x in a..b, |(\u230ax\u230b + 1 / 2 - x)| / x ^ (s+1).re := ZetaSum_aux1_4 apos a_lt_b\n    _ \u2264 \u222b x in a..b, 1 / x ^ (s.re + 1) := ZetaSum_aux1_5 apos a_lt_b \u03c3pos\n    _ = (a ^ (-s.re) - b ^ (-s.re)) / s.re := ?_\n  \u00b7 exact intervalIntegral.norm_integral_le_integral_norm (\u03bc := MeasureTheory.volume)\n      (a := a) (b := b) (f := fun x \u21a6 (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)) (le_of_lt a_lt_b)\n  \u00b7 refine ZetaSum_aux1_2 (c := s.re) apos a_lt_b \u27e8ne_of_gt \u03c3pos, ?_\u27e9\n    exact fun h \u21a6 (lt_self_iff_false 0).mp <| ZetaSum_aux1_1 apos a_lt_b h\n/-%%\n\\begin{proof}\\leanok\nApply the triangle inequality\n$$\n\\left|\\int_a^b \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\\right|\n\\le \\int_a^b \\frac{1}{x^{\\sigma+1}} \\, dx,\n$$\nand evaluate the integral.\n\\end{proof}\n%%-/\n\n-- no longer used\nlemma tsum_eq_partial_add_tail {N : \u2115} (f : \u2115 \u2192 \u2102) (hf : Summable f) :\n    \u2211' (n : \u2115), f n = (\u2211 n in Finset.Ico 0 N, f n) + \u2211' (n : \u2115), f (n + N) := by\n  rw [\u2190 sum_add_tsum_nat_add (f := f) (h := hf) (k := N), Finset.range_eq_Ico]\n\nlemma Finset.Ioc_eq_Ico (M N : \u2115): Finset.Ioc N M = Finset.Ico (N + 1) (M + 1) := by\n  ext a; simp only [Finset.mem_Ioc, Finset.mem_Ico]; constructor <;> intro \u27e8h\u2081, h\u2082\u27e9 <;> omega\n\nlemma finsetSum_tendsto_tsum {N : \u2115} {f : \u2115 \u2192 \u2102} (hf : Summable f) :\n    Tendsto (fun (k : \u2115) \u21a6 \u2211 n in Finset.Ioc N k, f n) atTop (\ud835\udcdd (\u2211' (n : \u2115), f (n + N))) := by\n  have := (Summable.hasSum_iff_tendsto_nat (f := fun n \u21a6 f (n + N))\n     (m := \u2211' (n : \u2115), f (n + N)) ?_).mp ?_\n  -- How to make the lengths of the intervals match?\n  \u00b7 convert this using 1 with M\n    ext M\n    rw [Finset.Ioc_eq_Ico, Finset.range_eq_Ico]\n    apply Finset.sum_equiv (g := fun n \u21a6 f (n + N)) ?_ ?_ ?_\n    \u00b7 sorry\n    \u00b7 sorry\n    \u00b7 sorry\n    -- Finset.sum_hom_rel\n  swap; apply (Summable.hasSum_iff ?_).mpr; rfl\n  all_goals exact summable_nat_add_iff N |>.mpr hf\n\nlemma tendsto_coe_atTop : Tendsto (fun (n : \u2115) \u21a6 (n : \u211d)) atTop atTop := by\n  rw [Filter.tendsto_atTop_atTop]\n  intro b\n  use \u230ab\u230b.toNat + 1\n  intro a ha\n  by_cases a_zero : a = 0\n  \u00b7 simp [a_zero] at ha\n  \u00b7 by_cases h : \u230ab\u230b.toNat < a\n    \u00b7 exact (Int.floor_lt.mp <| (Int.toNat_lt' a_zero).mp h).le\n    \u00b7 simp only [not_lt] at h\n      absurd le_trans ha h\n      simp\n\n-- related to `ArithmeticFunction.LSeriesSummable_zeta_iff.mpr s_re_gt`\nlemma Summable_rpow {s : \u2102} (s_re_gt : 1 < s.re) : Summable (fun (n : \u2115) \u21a6 1 / (n : \u2102) ^ s) := by\n  apply Summable.of_norm\n  have : s.re \u2260 0 := by linarith\n  simp only [one_div, norm_inv]\n  simp_rw [norm_natCast_cpow_of_re_ne_zero _ this]\n  exact (Real.summable_nat_rpow_inv (p := s.re)).mpr s_re_gt\n\nlemma Finset_coe_Nat_Int (f : \u2124 \u2192 \u2102) (m n : \u2115) :\n    (\u2211 x in Finset.Ioc m n, f x) = \u2211 x in Finset.Ioc (m : \u2124) n, f x := by\n/-\ninstead use `Finset.sum_map` and a version of `Nat.image_cast_int_Ioc` stated using `Finset.map`\n-/\n  apply Finset.sum_nbij (i := (fun (x : \u2115) \u21a6 (x : \u2124))) ?_ ?_ ?_ fun _ _ \u21a6 rfl\n  \u00b7 intro x hx; simp only [Finset.mem_Ioc, Nat.cast_lt, Nat.cast_le] at hx \u22a2; exact hx\n  \u00b7 intro x\u2081 _ x\u2082 _ h; simp only [Nat.cast_inj] at h; exact h\n  \u00b7 intro x hx\n    simp only [Finset.coe_Ioc, mem_image, mem_Ioc] at hx \u22a2\n    lift x to \u2115 using (by linarith); exact \u27e8x, by exact_mod_cast hx, rfl\u27e9\n\nlemma Complex.cpow_tendsto {s : \u2102} (s_re_gt : 1 < s.re) :\n    Tendsto (fun (x : \u2115) \u21a6 (x : \u2102) ^ (1 - s)) atTop (\ud835\udcdd 0) := by\n  have one_sub_s_re_ne : (1 - s).re \u2260 0 := by simp only [sub_re, one_re]; linarith\n  rw [tendsto_zero_iff_norm_tendsto_zero]\n  simp_rw [Complex.norm_natCast_cpow_of_re_ne_zero _ (one_sub_s_re_ne)]\n  rw [(by simp only [sub_re, one_re, neg_sub] : (1 - s).re = - (s - 1).re)]\n  apply (tendsto_rpow_neg_atTop _).comp tendsto_nat_cast_atTop_atTop; simp [s_re_gt]\n\nlemma Complex.cpow_inv_tendsto {s : \u2102} (hs : 0 < s.re) :\n    Tendsto (fun (x : \u2115) \u21a6 ((x : \u2102) ^ s)\u207b\u00b9) atTop (\ud835\udcdd 0) := by\n  rw [tendsto_zero_iff_norm_tendsto_zero]\n  simp_rw [norm_inv, Complex.norm_natCast_cpow_of_re_ne_zero _ <| ne_of_gt hs]\n  apply Filter.Tendsto.inv_tendsto_atTop\n  exact (tendsto_rpow_atTop hs).comp tendsto_nat_cast_atTop_atTop\n\nlemma ZetaSum_aux2a : \u2203 C, \u2200 (x : \u211d), |\u230ax\u230b + 1 / 2 - x| \u2264 C := by\n  use 1 / 2; exact ZetaSum_aux1_3\n\n/-%%\n\\begin{lemma}[ZetaSum_aux2]\\label{ZetaSum_aux2}\\lean{ZetaSum_aux2}\\leanok\n  Let $N$ be a natural number and $s\\in \\C$, $\\Re(s)>1$.\n  Then\n  \\[\n  \\sum_{N < n} \\frac{1}{n^s} =  \\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma ZetaSum_aux2 {N : \u2115} (N_pos : 0 < N) {s : \u2102} (s_re_gt : 1 < s.re) :\n    \u2211' (n : \u2115), 1 / (n + N : \u2102) ^ s =\n    (- N ^ (1 - s)) / (1 - s) - N ^ (-s) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1)) := by\n  have s_ne_zero : s \u2260 0 := fun hs \u21a6 by linarith [zero_re \u25b8 hs \u25b8 s_re_gt]\n  have s_ne_one : s \u2260 1 := fun hs \u21a6 (lt_self_iff_false _).mp <| one_re \u25b8 hs \u25b8 s_re_gt\n  apply tendsto_nhds_unique (X := \u2102) (Y := \u2115) (l := atTop)\n    (f := fun k \u21a6 ((k : \u2102) ^ (1 - s) - (N : \u2102) ^ (1 - s)) / (1 - s) + 1 / 2 * (1 / \u2191k ^ s) - 1 / 2 * (1 / \u2191N ^ s)\n      + s * \u222b (x : \u211d) in (N : \u211d)..k, (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1)))\n    (b := (- N ^ (1 - s)) / (1 - s) - N ^ (-s) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1)))\n  \u00b7 apply Filter.Tendsto.congr' (f\u2081 := fun (k : \u2115) \u21a6 \u2211 n in Finset.Ioc N k, 1 / (n : \u2102) ^ s) (l\u2081 := atTop)\n    \u00b7 apply Filter.eventually_atTop.mpr\n      use N + 1\n      intro k hk\n      convert ZetaSum_aux1 (a := N) (b := k) s_ne_one s_ne_zero \u27e8N_pos, hk\u27e9 using 1\n      convert Finset_coe_Nat_Int (fun n \u21a6 1 / (n : \u2102) ^ s) N k\n    \u00b7 convert finsetSum_tendsto_tsum (f := fun n \u21a6 1 / (n : \u2102) ^ s) (Summable_rpow s_re_gt); simp\n  \u00b7 apply (Tendsto.sub ?_ ?_).add (Tendsto.const_mul _ ?_)\n    \u00b7 rw [(by ring : -\u2191N ^ (1 - s) / (1 - s) = (0 - \u2191N ^ (1 - s)) / (1 - s) + 0)]\n      apply cpow_tendsto s_re_gt |>.sub_const _ |>.div_const _ |>.add\n      simp_rw [mul_comm_div, one_mul, one_div, (by congr; ring : \ud835\udcdd (0 : \u2102) = \ud835\udcdd ((0 : \u2102) / 2))]\n      apply Tendsto.div_const <| cpow_inv_tendsto (by positivity)\n    \u00b7 simp_rw [mul_comm_div, one_mul, one_div, cpow_neg]; exact tendsto_const_nhds\n    \u00b7 refine MeasureTheory.intervalIntegral_tendsto_integral_Ioi (a := N)\n        (b := (fun (n : \u2115) \u21a6 (n : \u211d))) ?_ tendsto_coe_atTop\n      apply MeasureTheory.Integrable.bdd_mul ?_ ?_\n      \u00b7 convert ZetaSum_aux2a; simp [\u2190 Complex.abs_ofReal]\n      \u00b7 apply integrableOn_Ioi_cpow_iff (by positivity) |>.mpr (by simp [s_re_gt]; positivity)\n      \u00b7 apply Measurable.aestronglyMeasurable\n        refine Measurable.sub (Measurable.add ?_ measurable_const) ?_\n        \u00b7 exact Measurable.comp (by exact fun _ _ \u21a6 trivial) Int.measurable_floor\n        \u00b7 exact Measurable.comp measurable_id measurable_ofReal\n/-%%\n\\begin{proof}\\uses{ZetaSum_aux1}\n  Apply Lemma \\ref{ZetaSum_aux1} with $a=N$ and $b\\to \\infty$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[ZetaBnd_aux1]\\label{ZetaBnd_aux1}\\lean{ZetaBnd_aux1}\\leanok\nFor any $N\\ge1$ and $s\\in \\C$, $\\sigma=\\Re(s)\\in(0,2]$,\n$$\n\\left| s\\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx \\right|\n\\ll |t| \\frac{N^{-\\sigma}}{\\sigma},\n$$\nas $|t|\\to\\infty$.\n\\end{lemma}\n%%-/\n\ndef ct_aux1 := 31381059610 -- 3 ^ 22 + 1\ndef C_aux1 := 100\n\nlemma ZetaBnd_aux1 (N : \u2115) (Npos : 1 \u2264 N) {\u03c3 : \u211d} (h\u03c3 : \u03c3 \u2208 Ioc 0 2) :\n    \u2200 (t : \u211d) (ht : ct_aux1 < |t|),\n    \u2016(\u03c3 + t * I) * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ ((\u03c3 + t * I) + 1)\u2016\n    \u2264 C_aux1 * |t| * N ^ (-\u03c3) / \u03c3 := by\n  have := @ZetaSum_aux1a (a := N)\n  sorry\n/-%%\n\\begin{proof}\\uses{ZetaSum_aux1a}\nApply Lemma \\ref{ZetaSum_aux1a} with $a=N$ and $b\\to \\infty$, and estimate $|s|\\ll |t|$.\n\\end{proof}\n%%-/\n\n\n/-%%\n\\begin{lemma}[HolomorphicOn_Zeta0]\\label{HolomorphicOn_Zeta0}\\lean{HolomorphicOn_Zeta0}\\leanok\nFor any $N\\ge1$, the function $\\zeta_0(N,s)$ is holomorphic on $\\{s\\in \\C\\mid \\Re(s)>0\\}$.\n\\end{lemma}\n%%-/\nlemma HolomorphicOn_riemannZeta0 {N : \u2115} (N_pos : 0 < N) :\n    HolomorphicOn (\u03b6\u2080 N) {s : \u2102 | s \u2260 1 \u2227 0 < s.re} := by\n  sorry\n/-%%\n\\begin{proof}\\uses{ZetaSum_aux1}\n  The function $\\zeta_0(N,s)$ is a finite sum of entire functions, plus an integral\n  that's absolutely convergent on $\\{s\\in \\C\\mid \\Re(s)>0 \u2227 s \\ne 1\\}$ by Lemma \\ref{ZetaSum_aux1}.\n%%-/\n\n-- MOVE TO MATHLIB near `differentiableAt_riemannZeta`\nlemma HolomophicOn_riemannZeta :\n    HolomorphicOn \u03b6 {s : \u2102 | s \u2260 1} := by\n  intro z hz\n  simp only [mem_setOf_eq] at hz\n  exact (differentiableAt_riemannZeta hz).differentiableWithinAt\n\n\n/-%%\n\\begin{lemma}[isPathConnected_aux]\\label{isPathConnected_aux}\\lean{isPathConnected_aux}\\leanok\nThe set $\\{s\\in \\C\\mid \\Re(s)>0 \u2227 s \\ne 1\\}$ is path-connected.\n\\end{lemma}\n%%-/\nlemma isPathConnected_aux : IsPathConnected {z : \u2102 | z \u2260 1 \u2227 0 < z.re} := by\n  use (2 : \u2102)\n  constructor; simp\n  intro y hy; simp only [ne_eq, mem_setOf_eq] at hy\n  by_cases h : y.re \u2264 1\n  \u00b7 apply JoinedIn.trans (y := I)\n    \u00b7 sorry\n    \u00b7 sorry\n  \u00b7 let f : \u211d \u2192 \u2102 := fun t \u21a6 y * t + 2 * (1 - t)\n    have cont : Continuous f := by continuity\n    apply JoinedIn.ofLine cont.continuousOn (by simp [f]) (by simp [f])\n    simp [f, unitInterval]\n    intro x hx; simp only [mem_Icc] at hx\n    simp only [mem_setOf_eq]\n    constructor\n    \u00b7 suffices \u00ac (2 - y) * x = 1 by\n        convert this using 1\n        ring_nf\n        sorry\n      simp [Complex.ext_iff]\n      contrapose!\n      intro hxy\n      rcases hxy with (hx1 | hy1)\n      \u00b7 have hyre: 2 - y.re < 1 := by linarith\n        by_cases hx2 : x = 0\n        \u00b7 simp only [hx2]; linarith\n        \u00b7 have := mul_lt_mul (a := 2 - y.re) (b := x) (c := 1) (d := 1) hyre hx.2\n            (lt_of_le_of_ne hx.1 <| ((Ne.def _ _).symm \u25b8 hx2).symm) (by norm_num)\n          linarith\n      \u00b7 simp [hy1]\n    \u00b7 sorry\n/-%%\n\\begin{proof}\n  Construct explicit paths from $2$ to any point, either a line segment or two joined ones.\n%%-/\n\n\n/-%%\n\\begin{lemma}[Zeta0EqZeta]\\label{Zeta0EqZeta}\\lean{Zeta0EqZeta}\\leanok\nFor $\\Re(s)>0$, $s\\ne1$, and for any $N$,\n$$\n\\zeta_0(N,s) = \\zeta(s).\n$$\n\\end{lemma}\n%%-/\nlemma Zeta0EqZeta {N : \u2115} (N_pos : 0 < N) {s : \u2102} (reS_pos : 0 < s.re) (s_ne_one : s \u2260 1) :\n    \u03b6\u2080 N s = riemannZeta s := by\n  let f := riemannZeta\n  let g := \u03b6\u2080 N\n  let U := {z : \u2102 | z \u2260 1 \u2227 0 < z.re}\n  have U_open : IsOpen U := by\n    refine IsOpen.inter isOpen_ne ?_\n    exact isOpen_lt (g := fun (z : \u2102) \u21a6 z.re) (by continuity) (by continuity)\n  have f_an : AnalyticOn \u2102 f U := by\n    apply (HolomophicOn_riemannZeta.analyticOn isOpen_ne).mono\n    simp only [ne_eq, setOf_subset_setOf, and_imp, U]\n    exact fun a ha _ \u21a6 ha\n  have g_an : AnalyticOn \u2102 g U := (HolomorphicOn_riemannZeta0 N_pos).analyticOn U_open\n  have preconU : IsPreconnected U := by\n    apply IsConnected.isPreconnected\n    apply (IsOpen.isConnected_iff_isPathConnected U_open).mp isPathConnected_aux\n  have h2 : (2 : \u2102) \u2208 U := by simp [U]\n  have s_mem : s \u2208 U := by simp [U, reS_pos, s_ne_one]\n  convert (AnalyticOn.eqOn_of_preconnected_of_eventuallyEq f_an g_an preconU h2 ?_ s_mem).symm\n  have u_mem : {z : \u2102 | 1 < z.re} \u2208 \ud835\udcdd (2 : \u2102) := by\n    apply mem_nhds_iff.mpr\n    use {z : \u2102 | 1 < z.re}\n    simp only [setOf_subset_setOf, imp_self, forall_const, mem_setOf_eq, re_ofNat,\n      Nat.one_lt_ofNat, and_true, true_and]\n    exact isOpen_lt (by continuity) (by continuity)\n  filter_upwards [u_mem]\n  intro z hz\n  simp only [f,g, zeta_eq_tsum_one_div_nat_cpow hz, riemannZeta0_apply]\n  nth_rewrite 2 [neg_div]\n  rw [\u2190 sub_eq_add_neg, \u2190 ZetaSum_aux2 N_pos hz, \u2190 sum_add_tsum_nat_add N (Summable_rpow hz)]\n  congr\n  simp\n/-%%\n\\begin{proof}\\leanok\n\\uses{ZetaSum_aux2, RiemannZeta0, HolomorphicOn_Zeta0, isPathConnected_aux}\nUse Lemma \\ref{ZetaSum_aux2} and the Definition \\ref{RiemannZeta0}.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[ZetaBnd_aux2]\\label{ZetaBnd_aux2}\\lean{ZetaBnd_aux2}\\leanok\nGiven $n \u2264 t$ and $\\sigma$ with $1-A/\\log t \\le \\sigma$, we have\nthat\n$$\n|n^{-s}| \\le n^{-1} e^A.\n$$\n\\end{lemma}\n%%-/\nlemma ZetaBnd_aux2 {n : \u2115} {t A \u03c3 : \u211d} (Apos : 0 < A) (\u03c3pos : 0 < \u03c3) (n_le_t : n \u2264 |t|)\n    (\u03c3_ge : (1 : \u211d) - A / |t|.log \u2264 \u03c3) :\n    \u2016(n : \u2102) ^ (-(\u03c3 + t * I))\u2016 \u2264 (n : \u211d)\u207b\u00b9 * Real.exp A := by\n  set s := \u03c3 + t * I\n  by_cases n0 : n = 0\n  \u00b7 simp_rw [n0, CharP.cast_eq_zero, inv_zero, zero_mul]\n    rw [Complex.zero_cpow ?_]; simp\n    exact fun h \u21a6 (NeZero.of_pos \u03c3pos).ne <| zero_eq_neg.mp <| zero_re \u25b8 h \u25b8 (by simp [s])\n  have n_gt_0 : 0 < n := Nat.pos_of_ne_zero n0\n  have n_gt_0' : (0 : \u211d) < (n : \u211d) := Nat.cast_pos.mpr n_gt_0\n  have n_ge_1 : 1 \u2264 (n : \u211d) := Nat.one_le_cast.mpr <| Nat.succ_le_of_lt n_gt_0\n  calc\n    _ = |((n : \u211d) ^ (-\u03c3))| := ?_\n    _ \u2264 Real.exp (Real.log n * -\u03c3) := Real.abs_rpow_le_exp_log_mul (n : \u211d) (-\u03c3)\n    _ \u2264 Real.exp (Real.log n *  -(1 - A / Real.log t)) := ?_\n    _ \u2264 Real.exp (- Real.log n + A) := Real.exp_le_exp_of_le ?_\n    _ \u2264 _ := by rw [Real.exp_add, Real.exp_neg, Real.exp_log n_gt_0']\n  \u00b7 have : \u2016(n : \u2102) ^ (-s)\u2016 = n ^ (-s.re) := abs_cpow_eq_rpow_re_of_pos n_gt_0' (-s)\n    rw [this, abs_eq_self.mpr <| Real.rpow_nonneg n_gt_0'.le _]; simp [s]\n  \u00b7 apply Real.exp_le_exp_of_le <| mul_le_mul_of_nonneg_left _ <| Real.log_nonneg n_ge_1\n    rw [neg_sub, neg_le_sub_iff_le_add, add_comm, \u2190 Real.log_abs]; linarith\n  \u00b7 simp only [neg_sub, le_neg_add_iff_add_le]\n    ring_nf\n    conv => rw [mul_comm, \u2190 mul_assoc, \u2190 Real.log_abs]; rhs; rw [\u2190 one_mul A]\n    gcongr\n    by_cases ht1 : |t| = 1; simp [ht1]\n    apply (inv_mul_le_iff ?_).mpr; convert Real.log_le_log n_gt_0' n_le_t using 1; rw [mul_one]\n    exact Real.log_pos <| lt_of_le_of_ne (le_trans n_ge_1 n_le_t) <| fun t \u21a6 ht1 (t.symm)\n/-%%\n\\begin{proof}\\leanok\nUse $|n^{-s}| = n^{-\\sigma}\n= e^{-\\sigma \\log n}\n\\le\n\\exp(-\\left(1-\\frac{A}{\\log t}\\right)\\log n)\n\\le\nn^{-1} e^A$,\nsince $n\\le t$.\n\\end{proof}\n%%-/\n\nlemma UpperBnd_aux {A \u03c3 t: \u211d} (A_pos : 0 < A) (A_lt : A < 1) (t_ge : 3 < |t|)\n      (\u03c3_ge : 1 - A / Real.log |t| \u2264 \u03c3) :\n      1 < Real.log |t| \u2227 1 - A < \u03c3 \u2227 0 < \u03c3 \u2227 \u03c3 + t * I \u2260 1:= by\n  have logt_gt_one: 1 < Real.log |t| := by\n    rw [\u2190 Real.log_exp (x := 1)]\n    apply Real.log_lt_log (Real.exp_pos _)\n    linarith [(by exact lt_trans Real.exp_one_lt_d9 (by norm_num) : Real.exp 1 < 3)]\n  have \u03c3_gt : 1 - A < \u03c3 := by\n    apply lt_of_lt_of_le ((sub_lt_sub_iff_left (a := 1)).mpr ?_) \u03c3_ge\n    exact (div_lt_iff (by linarith)).mpr <| lt_mul_right A_pos logt_gt_one\n  refine \u27e8logt_gt_one, \u03c3_gt, by linarith, ?__\u27e9\n  contrapose! t_ge\n  simp only [Complex.ext_iff, add_re, ofReal_re, mul_re, I_re, mul_zero, ofReal_im, I_im, mul_one,\n    sub_self, add_zero, one_re, add_im, mul_im, zero_add, one_im] at t_ge\n  norm_num [t_ge.2]\n\nlemma UpperBnd_aux2 {A \u03c3 t: \u211d} (A_pos : 0 < A) (A_lt : A < 1) (t_ge : 3 < |t|)\n      (\u03c3_ge : 1 - A / |t|.log \u2264 \u03c3) :\n      |t| ^ (1 - \u03c3) \u2264 A.exp := by\n  have : |t| ^ (1 - \u03c3) \u2264 |t| ^ (A / |t|.log) :=\n    Real.rpow_le_rpow_of_exponent_le (by linarith) (by linarith)\n  apply le_trans this ?_\n  conv => lhs; lhs; rw [\u2190 Real.exp_log (by linarith : 0 < |t|)]\n  rw [div_eq_mul_inv, Real.rpow_mul (by positivity), \u2190 Real.exp_mul, \u2190 Real.exp_mul, mul_comm,\n    \u2190 mul_assoc, inv_mul_cancel, one_mul]\n  apply Real.log_ne_zero.mpr; split_ands <;> linarith\n\nlemma riemannZeta0_zero_aux (N : \u2115) (Npos : 0 < N):\n    \u2211 x in Finset.Ico 0 N, ((x : \u211d))\u207b\u00b9 = \u2211 x in Finset.Ico 1 N, ((x : \u211d))\u207b\u00b9 := by\n  have : Finset.Ico 1 N \u2286 Finset.Ico 0 N := by\n    intro x hx\n    simp only [Finset.mem_Ico, Nat.Ico_zero_eq_range, Finset.mem_range] at hx \u22a2\n    exact hx.2\n  rw [\u2190 Finset.sum_sdiff (s\u2081 := Finset.Ico 1 N) (s\u2082 := Finset.Ico 0 N) this]\n  have : Finset.Ico 0 N \\ Finset.Ico 1 N = Finset.range 1 := by\n    ext a\n    simp only [Nat.Ico_zero_eq_range, Finset.mem_sdiff, Finset.mem_range, Finset.mem_Ico, not_and,\n      not_lt, Finset.range_one, Finset.mem_singleton]\n    constructor\n    \u00b7 intro \u27e8ha\u2081, ha\u2082\u27e9; omega\n    \u00b7 intro ha\n      constructor\n      \u00b7 simp [ha, Npos]\n      \u00b7 omega\n  rw [this]; simp\n\nlemma UpperBnd_aux3 {A C \u03c3 t : \u211d} (Apos : 0 < A) (A_lt_one : A < 1) {N : \u2115} (Npos : 0 < N)\n    (\u03c3_ge : 1 - A / Real.log |t| \u2264 \u03c3) (t_ge : 3 < |t|) (N_le_t : (N : \u211d) \u2264 |t|) (hC : 2 \u2264 C) :\n     \u2016\u2211 n in Finset.range N, (n : \u2102) ^ (-(\u03c3 + t * I))\u2016 \u2264 A.exp * C * |t|.log := by\n  obtain \u27e8logt_gt_one, _, \u03c3Pos, _\u27e9 := UpperBnd_aux Apos A_lt_one t_ge \u03c3_ge\n  have (n : \u2115) (hn : n \u2208 Finset.range N) := ZetaBnd_aux2 (n := n) Apos \u03c3Pos ?_ \u03c3_ge\n  swap; exact le_trans (Nat.cast_le.mpr (Finset.mem_range.mp hn).le) N_le_t\n  replace := norm_sum_le_of_le (Finset.range N) this\n  rw [\u2190 Finset.sum_mul, mul_comm _ A.exp] at this\n  rw [mul_assoc]\n  apply le_trans this <| (mul_le_mul_left A.exp_pos).mpr ?_\n  have : 1 + (N - 1: \u211d).log \u2264 C * |t|.log := by\n    by_cases hN : N = 1\n    \u00b7 simp only [hN, Nat.cast_one, sub_self, Real.log_zero, add_zero];\n      rw [\u2190 mul_one 1]; exact mul_le_mul (by linarith) logt_gt_one.le (by norm_num) (by positivity)\n    \u00b7 rw [(by ring : C * Real.log |t| = (C - 1) * Real.log |t| + Real.log |t|)]\n      replace hN : 0 < (N : \u211d) - 1 := by simp only [sub_pos, Nat.one_lt_cast]; omega\n      have : (N - 1: \u211d).log \u2264 |t|.log := Real.log_le_log hN (by linarith)\n      apply add_le_add ?_ this\n      nth_rewrite 1 [\u2190 mul_one 1]\n      exact mul_le_mul (by linarith) logt_gt_one.le (by norm_num) (by linarith)\n  refine le_trans ?_ this\n  convert harmonic_eq_sum_Icc \u25b8 harmonic_le_one_add_log (N - 1)\n  \u00b7 simp only [Rat.cast_sum, Rat.cast_inv, Rat.cast_natCast, Finset.range_eq_Ico]\n    rw [riemannZeta0_zero_aux N Npos]; congr! 1\n  \u00b7 rw [Nat.cast_pred Npos]\n\nlemma Nat.self_div_floor_bound {t : \u211d}  (t_ge : 1 \u2264 |t|) : (|t| / \u2191\u230a|t|\u230b\u208a) \u2208 Icc 1 2 := by\n  set N := \u230a|t|\u230b\u208a\n  have Npos : 0 < N := Nat.floor_pos.mpr (by linarith)\n  have N_le_t : N \u2264 |t| := by exact Nat.floor_le <| abs_nonneg _\n  constructor\n  \u00b7 apply le_div_iff (by simp [Npos]) |>.mpr; simp [N_le_t]\n  \u00b7 apply div_le_iff (by positivity) |>.mpr\n    suffices |t| < \u2191N + 1 by linarith [(by exact_mod_cast (by omega) : 1 \u2264 (N : \u211d))]\n    apply Nat.lt_floor_add_one\n\nlemma le_trans\u2084 {\u03b1 : Type*} [Preorder \u03b1] {a b c d: \u03b1} : a \u2264 b \u2192 b \u2264 c \u2192 c \u2264 d \u2192 a \u2264 d :=\n  fun hab hbc hcd \u21a6 le_trans (le_trans hab hbc) hcd\n\nlemma UpperBnd_aux5 {\u03c3 t : \u211d}  (t_ge : 3 < |t|) (\u03c3_le : \u03c3 \u2264 2) : (|t| / \u230a|t|\u230b\u208a) ^ \u03c3 \u2264 4 := by\n  obtain \u27e8h\u2081, h\u2082\u27e9 := Nat.self_div_floor_bound (by linarith)\n  refine le_trans\u2084 (c := 2 ^ 2) ?_ (Real.rpow_le_rpow (by linarith) h\u2082 (by norm_num)) (by norm_num)\n  exact (Real.rpow_le_rpow_of_exponent_le h\u2081 \u03c3_le)\n\nlemma UpperBnd_aux6 {\u03c3 t : \u211d} (t_ge : 3 < |t|) (\u03c3_gt : 1 / 2 < \u03c3) (\u03c3_le : \u03c3 \u2264 2)\n  (neOne : \u03c3 + t * I \u2260 1) (Npos : 0 < \u230a|t|\u230b\u208a) (N_le_t : \u230a|t|\u230b\u208a \u2264 |t|) :\n    \u230a|t|\u230b\u208a ^ (1 - \u03c3) / \u20161 - (\u03c3 + t * I)\u2016 \u2264 |t| ^ (1 - \u03c3) * 2 \u2227\n    \u230a|t|\u230b\u208a ^ (-\u03c3) / 2 \u2264 |t| ^ (1 - \u03c3) \u2227 \u230a|t|\u230b\u208a ^ (-\u03c3) / \u03c3 \u2264 8 * |t| ^ (-\u03c3) := by\n  have bnd := UpperBnd_aux5 t_ge \u03c3_le\n  have bnd' : (|t| / \u230a|t|\u230b\u208a) ^ \u03c3 \u2264 2 * |t| := by linarith\n  split_ands\n  \u00b7 apply (div_le_iff <| norm_pos_iff.mpr <| sub_ne_zero_of_ne neOne.symm).mpr\n    conv => rw [mul_assoc]; rhs; rw [mul_comm]\n    apply (div_le_iff <| Real.rpow_pos_of_pos (by linarith) _).mp\n    rw [div_rpow_eq_rpow_div_neg (by positivity) (by positivity), neg_sub]\n    refine le_trans\u2084 ?_ bnd' ?_\n    \u00b7 exact Real.rpow_le_rpow_of_exponent_le (one_le_div (by positivity) |>.mpr N_le_t) (by simp)\n    \u00b7 apply (mul_le_mul_left (by norm_num)).mpr; simpa using abs_im_le_abs (1 - (\u03c3 + t * I))\n  \u00b7 apply div_le_iff (by norm_num) |>.mpr\n    rw [Real.rpow_sub (by linarith), Real.rpow_one, div_mul_eq_mul_div, mul_comm]\n    apply div_le_iff (by positivity) |>.mp\n    convert bnd' using 1\n    rw [\u2190 Real.rpow_neg (by linarith), div_rpow_neg_eq_rpow_div (by positivity) (by positivity)]\n  \u00b7 apply div_le_iff (by positivity) |>.mpr\n    rw [mul_assoc, mul_comm, mul_assoc]\n    apply div_le_iff' (by positivity) |>.mp\n    apply le_trans ?_ (by linarith : 4 \u2264 \u03c3 * 8)\n    convert bnd using 1; exact div_rpow_neg_eq_rpow_div (by positivity) (by positivity)\n\nlemma norm_add\u2084_le {E: Type*} [SeminormedAddGroup E] (a : E) (b : E) (c : E) (d : E) :\n    \u2016a + b + c + d\u2016 \u2264 \u2016a\u2016 + \u2016b\u2016 + \u2016c\u2016 + \u2016d\u2016 := by\n  apply le_trans <| norm_add_le (a + b + c) d\n  simp only [add_le_add_iff_right]; apply norm_add\u2083_le\n\nlemma add_le_add_le_add {\u03b1 : Type*} [Add \u03b1] [Preorder \u03b1]\n    [CovariantClass \u03b1 \u03b1 (fun x x_1 \u21a6 x + x_1) fun x x_1 \u21a6 x \u2264 x_1]\n    [CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 \u21a6 x + x_1) fun x x_1 \u21a6 x \u2264 x_1]\n    {a b c d e f : \u03b1} (h\u2081 : a \u2264 b) (h\u2082 : c \u2264 d) (h\u2083 : e \u2264 f) : a + c + e \u2264 b + d + f :=\n  add_le_add (add_le_add h\u2081 h\u2082) h\u2083\n\nlemma add_le_add_le_add_le_add {\u03b1 : Type*} [Add \u03b1] [Preorder \u03b1]\n    [CovariantClass \u03b1 \u03b1 (fun x x_1 \u21a6 x + x_1) fun x x_1 \u21a6 x \u2264 x_1]\n    [CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 \u21a6 x + x_1) fun x x_1 \u21a6 x \u2264 x_1]\n    {a b c d e f g h : \u03b1} (h\u2081 : a \u2264 b) (h\u2082 : c \u2264 d) (h\u2083 : e \u2264 f) (h\u2084 : g \u2264 h) :\n    a + c + e + g \u2264 b + d + f + h:= add_le_add (add_le_add_le_add h\u2081 h\u2082 h\u2083) h\u2084\n\n/-%%\n\\begin{lemma}[ZetaUpperBnd]\\label{ZetaUpperBnd}\\lean{ZetaUpperBnd}\\leanok\nFor any $s\\in \\C$, $1/2 \\le \\Re(s)=\\sigma\\le 2$,\nand any $0 < A < 1$ sufficiently small, and $1-A/\\log |t| \\le \\sigma$, we have\n$$\n|\\zeta(s)| \\ll \\log t,\n$$\nas $|t|\\to\\infty$.\n\\end{lemma}\n%%-/\nlemma ZetaUpperBnd :\n    \u2203 (A : \u211d) (Apos : 0 < A) (C : \u211d) (Cpos : 0 < C), \u2200 (\u03c3 : \u211d) (t : \u211d) (t_ge : ct_aux1 < |t|)\n    (_ : \u03c3 \u2208 Icc (1 - A / |t|.log) 2), \u2016\u03b6 (\u03c3 + t * I)\u2016 \u2264 C * |t|.log := by\n  let A := (1 : \u211d) / 2\n  have Apos : 0 < A := by norm_num\n  let C := A.exp * (5 + 8 * C_aux1)\n  refine \u27e8A, Apos, C, (by positivity), ?_\u27e9\n  intro \u03c3 t t_ge \u27e8\u03c3_ge, \u03c3_le\u27e9\n  have t_ge' : 3 < |t| := lt_trans (by norm_num [ct_aux1]) t_ge\n  set N := \u230a|t|\u230b\u208a\n  have Npos : 0 < N := Nat.floor_pos.mpr (by linarith)\n  have N_le_t : N \u2264 |t| := Nat.floor_le <| abs_nonneg _\n  obtain \u27e8logt_gt_one, \u03c3_gt, \u03c3Pos, neOne\u27e9 := UpperBnd_aux Apos (by norm_num) t_ge' \u03c3_ge\n  norm_num [A] at \u03c3_gt\n  rw [\u2190 Zeta0EqZeta (N := N) Npos (by simp [\u03c3Pos]) neOne]\n  set s := \u03c3 + t * I\n  calc\n    _ \u2264 \u2016\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s\u2016 + \u2016(- N ^ (1 - s)) / (1 - s)\u2016 +\n      \u2016(-(N : \u2102) ^ (-s)) / 2\u2016 +\n      \u2016s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\u2016 := by apply norm_add\u2084_le\n    _ \u2264 A.exp * 2 * |t|.log + \u2016(- N ^ (1 - s)) / (1 - s)\u2016 + \u2016(-(N : \u2102) ^ (-s)) / 2\u2016 +\n      \u2016s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\u2016 := ?_\n    _ \u2264 A.exp * 2 * |t|.log + \u2016(- N ^ (1 - s)) / (1 - s)\u2016 + \u2016(-(N : \u2102) ^ (-s)) / 2\u2016 +\n      C_aux1 * |t| * N ^ (-\u03c3) / \u03c3  := ?_\n    _ = A.exp * 2 * |t|.log + N ^ (1 - \u03c3) / \u2016(1 - s)\u2016 + N ^ (-\u03c3) / 2 +\n      C_aux1 * |t| * N ^ (-\u03c3) / \u03c3  := ?_\n    _ \u2264 A.exp * 2 * |t|.log + |t| ^ (1 - \u03c3) * 2 +\n        |t| ^ (1 - \u03c3) + C_aux1 * |t| * (8 * |t| ^ (-\u03c3)) := ?_\n    _ = A.exp * 2 * |t|.log + (3 + 8 * C_aux1) * |t| ^ (1 - \u03c3) := ?_\n    _ \u2264 A.exp * 2 * |t|.log + (3 + 8 * C_aux1) * A.exp * 1 := ?_\n    _ \u2264 A.exp * 2 * |t|.log + (3 + 8 * C_aux1) * A.exp * |t|.log:= ?_\n    _ = _ := by ring\n  \u00b7 simp only [add_le_add_iff_right, one_div_cpow_eq_cpow_neg]\n    convert UpperBnd_aux3 (C := 2) Apos (by norm_num) Npos \u03c3_ge t_ge' N_le_t le_rfl\n  \u00b7 simp only [add_le_add_iff_left]; exact ZetaBnd_aux1 N (by linarith) \u27e8\u03c3Pos, \u03c3_le\u27e9 t t_ge\n  \u00b7 simp only [norm_div, norm_neg, norm_eq_abs, RCLike.norm_ofNat, Nat.abs_cast, s]\n    congr <;> (convert norm_natCast_cpow_of_pos Npos _; simp)\n  \u00b7 have \u27e8h\u2081, h\u2082, h\u2083\u27e9 := UpperBnd_aux6 t_ge' \u03c3_gt \u03c3_le neOne Npos N_le_t\n    refine add_le_add_le_add_le_add le_rfl h\u2081 h\u2082 ?_\n    rw [mul_div_assoc]\n    exact mul_le_mul_left (mul_pos (by norm_num [C_aux1]) (by positivity)) |>.mpr h\u2083\n  \u00b7 ring_nf; conv => lhs; rhs; lhs; rw [mul_assoc, mul_comm |t|]\n    rw [\u2190 Real.rpow_add_one (by positivity)]; ring_nf\n  \u00b7 simp only [Real.log_abs, add_le_add_iff_left, mul_one]\n    exact mul_le_mul_left (by positivity) |>.mpr <| UpperBnd_aux2 Apos (by norm_num) t_ge' \u03c3_ge\n  \u00b7 simp only [add_le_add_iff_left]\n    apply mul_le_mul_left (by norm_num [Real.exp_pos, C_aux1]) |>.mpr <| logt_gt_one.le\n/-%%\n\\begin{proof}\\uses{ZetaBnd_aux1, ZetaBnd_aux2, Zeta0EqZeta}\\leanok\nFirst replace $\\zeta(s)$ by $\\zeta_0(N,s)$ for $N = \\lfloor |t| \\rfloor$.\nWe estimate:\n$$\n|\\zeta_0(N,s)| \\ll\n\\sum_{1\\le n < |t|} |n^{-s}|\n+\n\\frac{- |t|^{1-\\sigma}}{|1-s|} + \\frac{-|t|^{-\\sigma}}{2} +\n|t| \\cdot |t| ^ (-\u03c3) / \u03c3\n$$\n$$\n\\ll\ne^A \\sum_{1\\le n < |t|} n^{-1}\n+|t|^{1-\\sigma}\n$$\n,\nwhere we used Lemma \\ref{ZetaBnd_aux2} and Lemma \\ref{ZetaBnd_aux1}.\nThe first term is $\\ll \\log |t|$.\nFor the second term, estimate\n$$\n|t|^{1-\\sigma}\n\\le |t|^{1-(1-A/\\log |t|)}\n= |t|^{A/\\log |t|} \\ll 1.\n$$\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[ZetaDerivUpperBnd]\\label{ZetaDerivUpperBnd}\\lean{ZetaDerivUpperBnd}\\leanok\nFor any $s\\in \\C$, $1/2 \\le \\Re(s)=\\sigma\\le 2$,\nthere is an $A>0$ so that for $1-A/\\log t \\le \\sigma$, we have\n$$\n|\\zeta'(s)| \\ll \\log^2 t,\n$$\nas $|t|\\to\\infty$.\n\\end{lemma}\n%%-/\nlemma ZetaDerivUpperBnd :\n    \u2203 (A : \u211d) (Apos : 0 < A) (C : \u211d) (Cpos : 0 < C), \u2200 (\u03c3 : \u211d) (t : \u211d) (t_gt : 3 < |t|)\n    (h\u03c3 : \u03c3 \u2208 Icc (1 - A / Real.log |t|) 2),\n    \u2016deriv \u03b6 (\u03c3 + t * I)\u2016 \u2264 C * (Real.log |t|) ^ 2 := by\n  let A := (1 : \u211d) / 2\n  have Apos : 0 < A := by norm_num\n  refine \u27e8A, Apos, 10, by norm_num, ?_\u27e9\n  intro \u03c3 t t_ge \u27e8\u03c3_ge, \u03c3_le\u27e9\n  set N := \u230a|t|\u230b\u208a\n  set s := \u03c3 + t * I\n  obtain \u27e8logt_gt_one, \u03c3_gt, \u03c3Pos, neOne\u27e9 := UpperBnd_aux Apos (by norm_num) t_ge \u03c3_ge\n  have : deriv \u03b6 s = deriv (\u03b6\u2080 N) s := by\n    have := Zeta0EqZeta (N := N) (Nat.floor_pos.mpr (by linarith)) (by simp [\u03c3Pos]) neOne\n    -- these functions agree on an open set, their derivatives agree there too\n    sorry\n  rw [this]\n  -- use calc similar to the one for ZetaUpperBnd\n  sorry\n/-%%\n\\begin{proof}\\uses{ZetaBnd_aux1, ZetaBnd_aux2, Zeta0EqZeta}\nFirst replace $\\zeta(s)$ by $\\zeta_0(N,s)$ for $N = \\lfloor |t| \\rfloor$.\nDifferentiating term by term, we get:\n$$\n\\zeta'(s) = -\\sum_{1\\le n < N} n^{-s} \\log n\n-\n\\frac{N^{1 - s}}{1 - s)^2} + \\frac{N^{1 - s} \\log N} {1 - s}\n+ \\frac{-N^{-s}\\log N}{2} +\n\\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n-\ns(s+1) \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+2}} \\, dx\n.\n$$\nEstimate as before, with an extra factor of $\\log |t|$.\n\\end{proof}\n%%-/\n\nlemma Tendsto_nhdsWithin_punctured_map_add {f : \u211d \u2192 \u211d} (a x : \u211d)\n    (f_mono : StrictMono f) (f_iso : Isometry f):\n    Tendsto (fun y \u21a6 f y + a) (\ud835\udcdd[>] x) (\ud835\udcdd[>] (f x + a)) := by\n  refine tendsto_iff_forall_eventually_mem.mpr ?_\n  intro v hv\n  simp only [mem_nhdsWithin] at hv\n  obtain \u27e8u, hu, hu2, hu3\u27e9 := hv\n  let t := {x | f x + a \u2208 u}\n  have : t \u2229 Ioi x \u2208 \ud835\udcdd[>] x := by\n    simp only [mem_nhdsWithin]\n    use t\n    simp only [subset_inter_iff, inter_subset_left, inter_subset_right, and_self,\n      and_true, t]\n    simp\n    refine \u27e8?_, by simp [hu2]\u27e9\n    simp [Metric.isOpen_iff] at hu \u22a2\n    intro x hx\n    obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 := hu (f x + a) hx\n    simp only [Metric.ball, dist_sub_eq_dist_add_right, setOf_subset_setOf] at h\u03b5 \u22a2\n    exact \u27e8\u03b5, \u03b5pos, fun _ hy \u21a6 h\u03b5 (by simp [isometry_iff_dist_eq.mp f_iso, hy])\u27e9\n  filter_upwards [this]\n  intro b hb\n  simp only [mem_inter_iff, mem_setOf_eq, mem_Ioi, t] at hb\n  refine hu3 ?_\n  simp only [mem_inter_iff, mem_Ioi, add_lt_add_iff_right]\n  exact \u27e8hb.1, f_mono hb.2\u27e9\n\nlemma Tendsto_nhdsWithin_punctured_add (a x : \u211d) :\n    Tendsto (fun y \u21a6 y + a) (\ud835\udcdd[>] x) (\ud835\udcdd[>] (x + a)) :=\n  Tendsto_nhdsWithin_punctured_map_add a x strictMono_id isometry_id\n\n/-%%\n\\begin{lemma}[ZetaNear1BndFilter]\\label{ZetaNear1BndFilter}\\lean{ZetaNear1BndFilter}\\leanok\nAs $\\sigma\\to1^+$,\n$$\n|\\zeta(\\sigma)| \\ll 1/(\\sigma-1).\n$$\n\\end{lemma}\n%%-/\n", "theoremStatement": "lemma ZetaNear1BndFilter:\n    (fun \u03c3 : \u211d \u21a6 \u03b6 \u03c3) =O[\ud835\udcdd[>](1 : \u211d)] (fun \u03c3 \u21a6 (1 : \u2102) / (\u03c3 - 1)) ", "theoremName": "ZetaNear1BndFilter", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "da0933e9736d1eb61b664799371adf15b27214cf", "date": "2024-04-15"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": "PrimeNumberTheoremAnd.ZetaBounds.jsonl", "positionMetadata": {"lineInFile": 1186, "tokenPositionInFile": 54152, "theoremPositionInFile": 84}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 80, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", 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"Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", 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"Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have := Tendsto_nhdsWithin_punctured_add (a := -1) (x := 1)\n  simp only [add_right_neg, \u2190 sub_eq_add_neg] at this\n  have := riemannZeta_isBigO_near_one_horizontal.comp_tendsto this\n  convert this using 1 <;> {ext; simp}", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 227}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module aux_results\n-/\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.Analysis.SpecialFunctions.Log.Basic\nimport Mathlib.Analysis.SpecialFunctions.NonIntegrable\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal\n\nnoncomputable section\n\nopen scoped BigOperators ArithmeticFunction\n\nopen Nat ArithmeticFunction Finset\n\n\nnamespace ArithmeticFunction.IsMultiplicative\n\nvariable {R : Type*}\n\ntheorem mult_lcm_eq_of_ne_zero [CommGroupWithZero R] (f : ArithmeticFunction R) (h_mult : f.IsMultiplicative) (x y : \u2115)\n    (hf : f (x.gcd y) \u2260 0) :\n    f (x.lcm y) = f x * f y / f (x.gcd y) := by\n  rw [\u2190h_mult.lcm_apply_mul_gcd_apply]\n  field_simp\n\ntheorem prod_factors_of_mult (f : ArithmeticFunction \u211d) (h_mult : ArithmeticFunction.IsMultiplicative f) {l : \u2115} (hl : Squarefree l) :\n    \u220f a : \u2115 in l.primeFactors, f a = f l := by\n  rw [\u2190IsMultiplicative.map_prod_of_subset_primeFactors h_mult l _ Finset.Subset.rfl,\n    Nat.prod_primeFactors_of_squarefree hl]\n\nend ArithmeticFunction.IsMultiplicative\n\nnamespace Aux\ntheorem sum_over_dvd_ite {\u03b1 : Type _} [Ring \u03b1] {P : \u2115} (hP : P \u2260 0) {n : \u2115} (hn : n \u2223 P)\n    {f : \u2115 \u2192 \u03b1} : \u2211 d in n.divisors, f d = \u2211 d in P.divisors, if d \u2223 n then f d else 0 :=\n  by\n  rw [\u2190Finset.sum_filter, Nat.divisors_filter_dvd_of_dvd hP hn]\n\n", "theoremStatement": "theorem sum_intro {\u03b1 M: Type _} [AddCommMonoid M] [DecidableEq \u03b1] (s : Finset \u03b1) {f : \u03b1 \u2192 M} (d : \u03b1)\n     (hd : d \u2208 s) :\n    f d = \u2211 k in s, if k = d then f k else 0 ", "theoremName": "Aux.sum_intro", "fileCreated": 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"Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  trans (\u2211 k in s, if k = d then f d else 0)\n  \u00b7 rw [sum_eq_single_of_mem d hd]\n    rw [if_pos rfl]\n    intro _ _ h; rw [if_neg h]\n  apply sum_congr rfl; intro k _; apply if_ctx_congr Iff.rfl _ (fun _ => rfl)\n  intro h; rw [h]", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 232}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Basic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.Wiener\n\nopen Asymptotics Complex ComplexConjugate Topology Filter Real MeasureTheory Set\n\nopen scoped Interval\n\n-- TODO: why do we need to bump this?\ninstance : MeasurableDiv\u2082 \u211d := by\n  haveI (G : Type) [DivInvMonoid G] [MeasurableSpace G] [MeasurableInv G] [MeasurableMul\u2082 G] :\n    MeasurableDiv\u2082 G := inferInstance\n  exact this \u211d\n\n/-%%\nIn this section, we prove the Perron formula, which plays a key role in our proof of Mellin inversion.\n%%-/\n\n/-%%\nThe following is preparatory material used in the proof of the Perron formula, see Lemma \\ref{formulaLtOne}.\n%%-/\n\n/-%\nTODO: move to general section.\n\\begin{lemma}[zeroTendstoDiff]\\label{zeroTendstoDiff}\\lean{zeroTendstoDiff}\\leanok\nIf the limit of $0$ is $L\u2081 - L\u2082$, then $L\u2081 = L\u2082$.\n\\end{lemma}\n%-/\nlemma zeroTendstoDiff (L\u2081 L\u2082 : \u2102) (f : \u211d \u2192 \u2102) (h : \u2200\u1da0 T in atTop,  f T = 0)\n    (h' : Tendsto f atTop (\ud835\udcdd (L\u2082 - L\u2081))) : L\u2081 = L\u2082 := by\n  rw [\u2190 zero_add L\u2081, \u2190 @eq_sub_iff_add_eq]\n  exact tendsto_nhds_unique (EventuallyEq.tendsto h) h'\n/-%\n\\begin{proof}\\leanok\nObvious.\n\\end{proof}\n%-/\n\n/-%\nTODO: Move this to general section.\n\\begin{lemma}[RectangleIntegral_tendsTo_VerticalIntegral]\\label{RectangleIntegral_tendsTo_VerticalIntegral}\\lean{RectangleIntegral_tendsTo_VerticalIntegral}\\leanok\n\\uses{RectangleIntegral}\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\lim_{T\\to\\infty}\\int_{\\sigma-iT}^{\\sigma'+iT}f(s)ds =\n\\int_{(\\sigma')}f(s)ds - \\int_{(\\sigma)}f(s)ds.$$\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_VerticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (T : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) atTop\n      (\ud835\udcdd (VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3)) := by\n/-%\n\\begin{proof}\\leanok\nAlmost by definition.\n%-/\n  simp only [RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im, ofReal_im,\n    mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add, zero_sub,\n    add_im]\n  apply Tendsto.sub\n  \u00b7 rewrite [\u2190 zero_add (VerticalIntegral _ _), \u2190 zero_sub_zero]\n    apply Tendsto.add <| Tendsto.sub (hbot.comp tendsto_neg_atTop_atBot) htop\n    exact (intervalIntegral_tendsto_integral hright tendsto_neg_atTop_atBot tendsto_id).const_smul I\n  \u00b7 exact (intervalIntegral_tendsto_integral hleft tendsto_neg_atTop_atBot tendsto_id).const_smul I\n--%\\end{proof}\n\nlemma verticalIntegral_eq_verticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hf : HolomorphicOn f ([[\u03c3,  \u03c3']] \u00d7\u2102 univ))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    VerticalIntegral f \u03c3 = VerticalIntegral f \u03c3' := by\n  refine zeroTendstoDiff _ _ _ (univ_mem' fun _ \u21a6 ?_)\n    (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  exact integral_boundary_rect_eq_zero_of_differentiableOn f _ _\n    (hf.mono fun z hrect \u21a6 \u27e8by simpa using hrect.1, trivial\u27e9)\n\nlemma verticalIntegral_sub_verticalIntegral_eq_squareIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102} {p : \u2102}\n    (h\u03c3: \u03c3 < p.re \u2227 p.re < \u03c3') (hf : HolomorphicOn f (Icc \u03c3  \u03c3' \u00d7\u2102 univ \\ {p}))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3 =\n    RectangleIntegral f (-c - c * I + p) (c + c * I + p) := by\n  have : Icc \u03c3 \u03c3' \u00d7\u2102 univ \u2208 \ud835\udcdd p := by\n    rw [\u2190 mem_interior_iff_mem_nhds, Complex.interior_reProdIm, interior_Icc, interior_univ]\n    refine \u27e8\u27e8?_, ?_\u27e9, trivial\u27e9 <;> linarith\n  obtain \u27e8c', hc'0, hc'\u27e9 := ((nhds_hasBasis_square p).1 _).mp this\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' hc'0] with c \u27e8hc0, hcc'\u27e9\n  have hsub : Square p c \u2286 Icc \u03c3 \u03c3' \u00d7\u2102 univ := (square_subset_square hc0 hcc'.le).trans hc'\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  apply Filter.EventuallyEq.tendsto\n  filter_upwards [Filter.Ioi_mem_atTop ((c - p.im) \u2294 (c + p.im))] with y hy\n  have : c - p.im < y \u2227 c + p.im < y := sup_lt_iff.mp hy\n  have : c + \u03c3 \u2264 p.re := by simpa using (hsub \u27e8left_mem_uIcc, left_mem_uIcc\u27e9).1.1\n  have : c + p.re \u2264 \u03c3' := by simpa using (hsub \u27e8right_mem_uIcc, right_mem_uIcc\u27e9).1.2\n  apply RectanglePullToNhdOfPole'\n  \u00b7 simpa using \u27e8by linarith, by linarith, by linarith\u27e9\n  \u00b7 exact square_mem_nhds p (ne_of_gt hc0)\n  \u00b7 apply RectSubRect' <;> simpa using by linarith\n  \u00b7 refine hf.mono (diff_subset_diff ?_ subset_rfl)\n    simpa [Rectangle, uIcc_of_lt (h\u03c3.1.trans h\u03c3.2)] using fun x \u27e8hx, _\u27e9 \u21a6 \u27e8hx, trivial\u27e9\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_UpperU]\\label{RectangleIntegral_tendsTo_UpperU}\\lean{RectangleIntegral_tendsTo_UpperU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma+iT}^{\\sigma'+iU}f(s)ds$$\nas $U\\to\\infty$ is the ``UpperUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_UpperU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 + I * T) (\u03c3' + I * U)) atTop\n      (\ud835\udcdd (UpperUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, UpperUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  + I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  + I * t).im = t  := by simp\n  have hbot : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) := by\n    exact tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in T..U, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Ioi T, f (s + y * I)) := by\n    exact (intervalIntegral_tendsto_integral_Ioi T int.restrict tendsto_id).const_smul I\n  have := ((hbot.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  simpa only [RectangleIntegral, UpperUIntegral, h_re, h_im, sub_zero, \u2190integral_Ici_eq_integral_Ioi]\n--%\\end{proof}\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_LowerU]\\label{RectangleIntegral_tendsTo_LowerU}\\lean{RectangleIntegral_tendsTo_LowerU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to -\\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma-iU}^{\\sigma'-iT}f(s)ds$$\nas $U\\to\\infty$ is the ``LowerUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_LowerU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * U) (\u03c3' - I * T)) atTop\n      (\ud835\udcdd (- LowerUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, LowerUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  - I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  - I * t).im = -t  := by simp\n  have hbot' : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - y * I)) atTop (\ud835\udcdd 0) := by\n    convert (hbot.comp tendsto_neg_atTop_atBot) using 1\n    ext; simp only [Function.comp_apply, ofReal_neg, neg_mul]; rfl\n  have htop : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) :=\n    tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in -U..-T, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Iic (-T), f (s + y * I)) := by\n    have := (intervalIntegral_tendsto_integral_Iic (-T) int.restrict tendsto_id).const_smul I\n    convert (this.comp tendsto_neg_atTop_atBot) using 1\n  have := ((hbot'.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  rw [zero_sub] at this\n  simp_rw [RectangleIntegral, LowerUIntegral, HIntegral, VIntegral, h_re, h_im, ofReal_neg, neg_mul, neg_add_rev, neg_sub]\n  have final : (((-\u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I)) + I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) -\n      I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) = (-(I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) +\n      ((I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) - \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I))) := by\n    ring_nf\n  exact final \u25b8 this\n--%\\end{proof}\n\n/-%%\nTODO : Move to general section\n\\begin{lemma}[limitOfConstant]\\label{limitOfConstant}\\lean{limitOfConstant}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma>0$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstant {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : 0 < \u03c3') (_ : 0 < \u03c3''), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atTop (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\\begin{align*}\n\\lim_{\\sigma'\\to\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atTop \u03c3) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 (\u03c3pos.trans_le h) \u03c3pos\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[limitOfConstantLeft]\\label{limitOfConstantLeft}\\lean{limitOfConstantLeft}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma<-3/2$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to-\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstantLeft {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3lt : \u03c3 \u2264 -3/2)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : \u03c3' \u2264 -3/2) (_ : \u03c3'' \u2264 -3/2), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atBot (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\begin{align*}\n\\lim_{\\sigma'\\to-\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to-\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atBot (-3/2)) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 h \u03c3lt\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_lt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\leanok\nLet $x>0$ and $x<1$. Then\n$$\\lim_{\\sigma\\to\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_lt_one {x : \u211d}  (xpos : 0 < x) (x_lt_one : x < 1) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nStandard.\n%%-/\n  have := Tendsto.mul_const C (tendsto_rpow_atTop_of_base_lt_one x (by linarith) x_lt_one)\n  simpa only [rpow_eq_pow, zero_mul] using this\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_gt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\leanok\nLet $x>1$. Then\n$$\\lim_{\\sigma\\to-\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_gt_one {x : \u211d} (x_gt_one : 1 < x) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atBot (\ud835\udcdd 0) := by\n  have := (zero_lt_one.trans x_gt_one)\n  have h := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one (inv_pos.mpr this) (inv_lt_one x_gt_one) C\n  convert (h.comp tendsto_neg_atBot_atTop) using 1\n  ext; simp only [this.le, inv_rpow, Function.comp_apply, rpow_neg, inv_inv]\n\n/-%%\n\\begin{proof}\\leanok\nStandard.\n\\end{proof}\n%%-/\n\n-- TODO: move near `Complex.cpow_neg`?\nlemma Complex.cpow_inv_ofReal_pos {a : \u211d} (ha : 0 \u2264 a) (r : \u2102) :\n    ((a : \u2102) ^ r)\u207b\u00b9 = (a : \u2102)\u207b\u00b9 ^ r := by\n  sorry\n\nlemma Complex.cpow_eq_exp_log_ofReal (x : \u211d) (hx : 0 < x) (y : \u2102) :\n    (x : \u2102) ^ y = Complex.exp (Real.log x * y) := by\n  simp [\u2190 Complex.cpow_eq_pow, Complex.cpow, hx.ne.symm, \u2190 Complex.ofReal_log hx.le]\n\n-- TODO: move near `Complex.mul_cpow_ofReal_nonneg`\nlemma Complex.cpow_neg_eq_inv_pow_ofReal_pos {a : \u211d} (ha : 0 < a) (r : \u2102) :\n    (a : \u2102) ^ (-r) = (a\u207b\u00b9 : \u2102) ^ r := by\n  rw [cpow_neg, \u2190 Complex.inv_cpow]\n  exact slitPlane_arg_ne_pi (Or.inl ha)\n\nnamespace Perron\n\nvariable {x \u03c3 \u03c3' \u03c3'' T : \u211d}\n\nnoncomputable abbrev f (x : \u211d) := fun (s : \u2102) \u21a6 x ^ s / (s * (s + 1))\n\n\nlemma f_mul_eq_f {x t : \u211d} (tpos : 0 < t) (xpos : 0 < x) (s : \u2102) : f t s * (x : \u2102) ^ (-s) = f (t / x) s := by\n  by_cases s_eq_zero : s = 0\n  \u00b7 simp [f, s_eq_zero]\n  by_cases s_eq_neg_one : s = -1\n  \u00b7 simp [f, s_eq_neg_one]\n  field_simp [f, mul_ne_zero s_eq_zero (fun hs \u21a6 add_eq_zero_iff_eq_neg.mp hs |> s_eq_neg_one)]\n  convert (Complex.mul_cpow_ofReal_nonneg tpos.le (inv_pos.mpr xpos).le s).symm using 2\n  \u00b7 convert Complex.cpow_neg_eq_inv_pow_ofReal_pos xpos s\n    exact ofReal_inv x\n  \u00b7 simp only [ofReal_inv]; rfl\n\n/-%%\n\\begin{lemma}[isHolomorphicOn]\\label{isHolomorphicOn}\\lean{Perron.isHolomorphicOn}\\leanok\nLet $x>0$. Then the function $f(s) = x^s/(s(s+1))$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n\\end{lemma}\n%%-/\nlemma isHolomorphicOn (xpos : 0 < x) : HolomorphicOn (f x) {0, -1}\u1d9c := by\n/-%%\n\\begin{proof}\\leanok\nComposition of differentiabilities.\n%%-/\n  unfold f\n  simp_rw [Complex.cpow_def_of_ne_zero <| ofReal_ne_zero.mpr <| ne_of_gt xpos]\n  apply DifferentiableOn.div <| DifferentiableOn.cexp <| DifferentiableOn.const_mul differentiableOn_id _\n  \u00b7 exact DifferentiableOn.mul differentiableOn_id <| DifferentiableOn.add_const differentiableOn_id 1\n  \u00b7 intro x hx\n    obtain \u27e8h0, h1\u27e9 := not_or.mp hx\n    exact mul_ne_zero h0 <| add_ne_add_left 1 |>.mpr h1 |>.trans_eq (add_left_neg 1)\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[integralPosAux]\\label{integralPosAux}\\lean{Perron.integralPosAux}\\leanok\nThe integral\n$$\\int_\\R\\frac{1}{|(1+t^2)(2+t^2)|^{1/2}}dt$$\nis positive (and hence convergent - since a divergent integral is zero in Lean, by definition).\n\\end{lemma}\n%%-/\n\nlemma integral_one_div_const_add_sq_pos (c : \u211d) (hc : 0 < c) : 0 < \u222b (t : \u211d), 1 / (c + t ^ 2) := by\n  have hfun_eq (t : \u211d) : 1 / (c + t ^ 2) = c\u207b\u00b9 * (1 + (c.sqrt\u207b\u00b9 * t) ^ 2)\u207b\u00b9 := by\n    field_simp [hc.ne.symm]\n  simp_rw [hfun_eq, MeasureTheory.integral_mul_left,\n    Measure.integral_comp_mul_left (fun t \u21a6 (1 + t ^ 2)\u207b\u00b9) (a:=c.sqrt\u207b\u00b9)]\n  simp [abs_eq_self.mpr <| Real.sqrt_nonneg c,\n    mul_pos (inv_pos.mpr hc) <| mul_pos (sqrt_pos.mpr hc) Real.pi_pos]\n\nlemma Integrable.one_div_const_add_sq (c : \u211d) (hc : 0 < c) : Integrable fun (t : \u211d) \u21a6 1 / (c + t ^ 2) :=\n  .of_integral_ne_zero (integral_one_div_const_add_sq_pos c hc).ne'\n\nlemma integralPosAux'_of_le (c\u2081 c\u2082 : \u211d) (c\u2081_pos : 0 < c\u2081) (hle : c\u2081 \u2264 c\u2082) :\n    0 < \u222b (t : \u211d), 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) := by\n  have c\u2082_pos : 0 < c\u2082 := by linarith\n  have hlower (t : \u211d) : 1 / (c\u2082 + t ^ 2) \u2264 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) := by\n    gcongr\n    calc\n      _ \u2264 (c\u2082 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt := by gcongr; apply Real.sqrt_le_sqrt; gcongr\n      _ \u2264 c\u2082 + t ^ 2 := by rw [\u2190 Real.sqrt_mul, sqrt_mul_self] <;> positivity\n  have hupper (t : \u211d) : 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) \u2264 1 / (c\u2081 + t ^ 2)  := by\n      gcongr\n      calc\n        _ \u2265 (c\u2081 + t ^ 2).sqrt * (c\u2081 + t ^ 2).sqrt := by gcongr; apply Real.sqrt_le_sqrt; gcongr\n        _ \u2265 c\u2081 + t ^ 2 := by rw [\u2190 Real.sqrt_mul, sqrt_mul_self] <;> positivity\n  calc 0 < \u222b t, 1 / (c\u2082 + t^2) := integral_one_div_const_add_sq_pos c\u2082 c\u2082_pos\n       _ \u2264 \u222b t, 1 / (Real.sqrt (c\u2081 + t^2) * Real.sqrt (c\u2082 + t^2)) := ?_\n  refine integral_mono (Integrable.one_div_const_add_sq c\u2082 c\u2082_pos) ?_ hlower\n  apply MeasureTheory.Integrable.mono (g := fun t:\u211d \u21a6 1/(c\u2081 + t^2)) <| Integrable.one_div_const_add_sq c\u2081 c\u2081_pos\n  \u00b7 refine (measurable_const.div <| Measurable.mul ?_ ?_).aestronglyMeasurable <;>\n      exact (measurable_const.add <| measurable_id'.pow_const 2).sqrt\n  \u00b7 refine ae_of_all _ (fun x \u21a6 ?_)\n    repeat rewrite [norm_of_nonneg (by positivity)]\n    exact hupper x\n\n\nlemma integralPosAux' (c\u2081 c\u2082 : \u211d) (c\u2081_pos : 0 < c\u2081) (c\u2082_pos : 0 < c\u2082) :\n    0 < \u222b (t : \u211d), 1 / ((c\u2081 + t^2).sqrt * (c\u2082 + t^2).sqrt) := by\n  by_cases hc : c\u2081 \u2264 c\u2082\n  \u00b7 exact integralPosAux'_of_le c\u2081 c\u2082 c\u2081_pos hc\n  \u00b7 convert integralPosAux'_of_le c\u2082 c\u2081 c\u2082_pos (by linarith) using 4; rw [mul_comm]\n\nlemma integralPosAux : 0 < \u222b (t : \u211d), 1 / ((1 + t^2).sqrt * (2 + t^2).sqrt) := by\n/-%%\n\\begin{proof}\\leanok\nThis integral is between $\\frac{1}{2}$ and $1$ of the integral of $\\frac{1}{1+t^2}$, which is $\\pi$.\n%%-/\n  apply integralPosAux' <;> norm_num\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[vertIntBound]\\label{vertIntBound}\\lean{Perron.vertIntBound}\\leanok\nLet $x>0$ and $\\sigma>1$. Then\n$$\\left|\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds\\right| \\leq x^\\sigma \\int_\\R\\frac{1}{|(1+t ^ 2)(2+t ^ 2)|^{1/2}}dt.$$\n\\end{lemma}\n%%-/\nlemma vertIntBound (xpos : 0 < x) (\u03c3_gt_one : 1 < \u03c3) :\n    \u2016VerticalIntegral (f x) \u03c3\u2016 \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt) := by\n  calc\n    _ = \u2016\u222b (t : \u211d), x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ \u2264 \u222b (t : \u211d), \u2016x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 :=\n        norm_integral_le_integral_norm _\n    _ = \u222b (t : \u211d), x ^ \u03c3 / \u2016((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ = x ^ \u03c3 * \u222b (t : \u211d), 1 / (\u2016\u03c3 + t * I\u2016 * \u2016\u03c3 + t * I + 1\u2016) := ?_\n    _ \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt) :=\n        mul_le_mul_of_nonneg_left ?_ (rpow_nonneg xpos.le _)\n  \u00b7 simp [VerticalIntegral]\n  \u00b7 simp [Complex.abs_cpow_eq_rpow_re_of_pos xpos]\n  \u00b7 simp [integral_mul_left, div_eq_mul_inv]\n  by_cases hint : Integrable fun (a : \u211d) \u21a6 1 / (\u2016\u03c3 + a * I\u2016 * \u2016\u03c3 + a * I + 1\u2016)\n  swap; rw [integral_undef hint]; exact integral_nonneg <| fun t \u21a6 by positivity\n  conv => rhs; rhs; intro a; rhs\n  apply integral_mono hint\n  \u00b7 have := integralPosAux\n    contrapose! this\n    simp_rw [integral_undef this, le_rfl]\n  rw [Pi.le_def]\n  intro t\n  gcongr <;> apply sqrt_le_sqrt\n  \u00b7 simp_rw [normSq_add_mul_I, add_le_add_iff_right, one_le_pow_of_one_le \u03c3_gt_one.le _]\n  \u00b7 rw [add_right_comm, \u2190 ofReal_one, \u2190 ofReal_add, normSq_add_mul_I, add_le_add_iff_right]\n    nlinarith\n  rfl\n/-%%\n\\begin{proof}\\leanok\n\\uses{VerticalIntegral}\nTriangle inequality and pointwise estimate.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[vertIntBoundLeft]\\label{vertIntBoundLeft}\\lean{Perron.vertIntBoundLeft}\\leanok\nLet $x>1$ and $\\sigma<-3/2$. Then\n$$\\left|\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds\\right| \\leq x^\\sigma \\int_\\R\\frac{1}{|(1/4+t ^ 2)(2+t ^ 2)|^{1/2}}dt.$$\n\\end{lemma}\n%%-/\n\nlemma vertIntBoundLeft (xpos : 0 < x) :\n    \u2203 C, \u2200 (\u03c3 : \u211d) (_ : \u03c3 < -3 / 2), \u2016VerticalIntegral' (f x) \u03c3\u2016 \u2264 C * x ^ \u03c3 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{VerticalIntegral}\n%%-/\n  /- This proof is adapted from `vertIntBound` -/\n  use 1 / (2 * \u03c0) *  \u2016(\u222b (t : \u211d), 1 / ((4\u207b\u00b9 + t ^ 2).sqrt * (4\u207b\u00b9 + t ^ 2).sqrt : \u2102))\u2016\n  intro \u03c3 h\u03c3\n  simp only [VerticalIntegral', abs_of_pos Real.pi_pos, smul_eq_mul, norm_mul, f]\n  rw [(by simp [pi_nonneg] : \u20161 / (2 * \u2191\u03c0 * I)\u2016 = 1 / (2 * \u03c0)), mul_assoc]\n  apply (mul_le_mul_left (by simp [pi_pos])).mpr\n  calc\n    _ = \u2016\u222b (t : \u211d), x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ \u2264 \u222b (t : \u211d), \u2016x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := norm_integral_le_integral_norm _\n    _ = \u222b (t : \u211d), x ^ \u03c3 / \u2016((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ = x ^ \u03c3 * \u222b (t : \u211d), 1 / (\u2016\u03c3 + t * I\u2016 * \u2016\u03c3 + t * I + 1\u2016) := ?_\n    _ \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((4\u207b\u00b9 + t ^ 2).sqrt * (4\u207b\u00b9 + t ^ 2).sqrt) := ?_\n    _ \u2264 _ := ?_\n  \u00b7 simp [VerticalIntegral, Real.pi_nonneg]\n  \u00b7 congr with t\n    rw [norm_div, Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos xpos, add_re, ofReal_re,\n      re_ofReal_mul, I_re, mul_zero, add_zero]\n  \u00b7 simp_rw [div_eq_mul_inv, integral_mul_left, one_mul, Complex.norm_eq_abs, map_mul]\n  \u00b7 gcongr\n    by_cases hint : Integrable fun (a : \u211d) \u21a6 1 / (\u2016\u03c3 + \u2191a * I\u2016 * \u2016\u03c3 + \u2191a * I + 1\u2016)\n    swap\n    \u00b7 rw [integral_undef hint]\n      exact integral_nonneg <| fun t \u21a6 by simp only [Pi.le_def, Pi.zero_apply]; positivity\n    apply integral_mono hint\n    \u00b7 have := integralPosAux' (4\u207b\u00b9) (4\u207b\u00b9) (by norm_num) (by norm_num)\n      contrapose! this\n      simp_rw [integral_undef this, le_rfl]\n    rw [Pi.le_def]\n    intro t\n    gcongr <;> apply sqrt_le_sqrt\n    \u00b7 rw [normSq_add_mul_I, add_le_add_iff_right]; ring_nf; nlinarith\n    \u00b7 rw [(by push_cast; ring : \u03c3 + t * I + 1 = ofReal' (\u03c3 + 1) + t * I),\n        normSq_add_mul_I, add_le_add_iff_right]; ring_nf; nlinarith\n  \u00b7 rw [mul_comm]\n    gcongr\n    \u00b7 have : 0 \u2264 \u222b (t : \u211d), 1 / (sqrt (4\u207b\u00b9 + t ^ 2) * sqrt (4\u207b\u00b9 + t ^ 2)) := by positivity\n      rw [\u2190 _root_.abs_of_nonneg this, \u2190 Complex.abs_ofReal]\n      apply le_of_eq; congr; norm_cast; exact integral_ofReal.symm\n/-%%\nTriangle inequality and pointwise estimate.\n\\end{proof}\n%%-/\n\nlemma map_conj (hx : 0 \u2264 x) (s : \u2102) : f x (conj s) = conj (f x s) := by\n  simp only [f, map_div\u2080, map_mul, map_add, map_one]\n  congr\n  rw [cpow_conj, Complex.conj_ofReal]; rw [Complex.arg_ofReal_of_nonneg hx]; exact pi_ne_zero.symm\n\ntheorem isTheta_uniformlyOn_uIcc {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    (fun (\u03c3, (y : \u211d)) \u21a6 f x (\u03c3 + y * I)) =\u0398[\ud835\udcdf [[\u03c3', \u03c3'']] \u00d7\u02e2 (atBot \u2294 atTop)]\n    ((fun y \u21a6 1 / y^2) \u2218 Prod.snd) := by\n  set l := \ud835\udcdf [[\u03c3', \u03c3'']] \u00d7\u02e2 (atBot \u2294 atTop : Filter \u211d) with hl\n  refine IsTheta.div (isTheta_norm_left.mp ?_) ?_\n  \u00b7 suffices (fun (\u03c3, _y) \u21a6 |x| ^ \u03c3) =\u0398[l] fun _ \u21a6 (1 : \u211d) by\n      simpa [Complex.abs_cpow_of_ne_zero <| ofReal_ne_zero.mpr (ne_of_gt xpos),\n        arg_ofReal_of_nonneg xpos.le] using this\n    exact (continuousOn_const.rpow continuousOn_id fun _ _ \u21a6 Or.inl <| ne_of_gt (abs_pos_of_pos xpos))\n      |>.const_isThetaUniformlyOn_isCompact isCompact_uIcc (by norm_num)\n      (fun i _ \u21a6 ne_of_gt <| rpow_pos_of_pos (abs_pos_of_pos xpos) _) _\n  \u00b7 have h_c {c : \u2102} : (fun (_ : \u211d \u00d7 \u211d) \u21a6 c) =o[l] Prod.snd := by\n      rewrite [hl, Filter.prod_sup, isLittleO_sup]\n      exact \u27e8isLittleO_const_snd_atBot c _, isLittleO_const_snd_atTop c _\u27e9\n    have h_yI : (fun ((_\u03c3, y) : \u211d \u00d7 \u211d) \u21a6 y * I) =\u0398[l] Prod.snd :=\n      isTheta_of_norm_eventuallyEq (by simp)\n    have h_\u03c3_yI : (fun (\u03c3y : \u211d \u00d7 \u211d) \u21a6 \u03c3y.1 + \u03c3y.2 * I) =\u0398[l] Prod.snd := by\n      refine IsLittleO.add_isTheta ?_ h_yI\n      exact continuous_ofReal.continuousOn.const_isBigOUniformlyOn_isCompact isCompact_uIcc\n        (by norm_num : \u2016(1 : \u2102)\u2016 \u2260 0) _ |>.trans_isLittleO h_c\n    simp_rw [sq]; exact h_\u03c3_yI.mul (h_\u03c3_yI.add_isLittleO h_c)\n\ntheorem isTheta_uniformlyOn_uIoc {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    (fun (\u03c3, (y : \u211d)) \u21a6 f x (\u03c3 + y * I)) =\u0398[\ud835\udcdf (uIoc \u03c3' \u03c3'') \u00d7\u02e2 (atBot \u2294 atTop)]\n    fun (\u03c3, y) \u21a6 1 / y^2 := by\n  refine (\ud835\udcdf (uIoc \u03c3' \u03c3'')).eq_or_neBot.casesOn (fun hbot \u21a6 by simp [hbot]) (fun _ \u21a6 ?_)\n  haveI : NeBot (atBot (\u03b1 := \u211d) \u2294 atTop) := sup_neBot.mpr (Or.inl atBot_neBot)\n  exact (isTheta_uniformlyOn_uIcc xpos \u03c3' \u03c3'').mono (by simpa using Ioc_subset_Icc_self)\n\nlemma isTheta (xpos : 0 < x) :\n    ((fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) =\u0398[atBot] fun (y : \u211d) \u21a6 1 / y^2) \u2227\n    (fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) =\u0398[atTop] fun (y : \u211d) \u21a6 1 / y^2 :=\n  isTheta_sup.mp <| isTheta_of_isThetaUniformly (isTheta_uniformlyOn_uIcc xpos \u03c3 \u03c3) left_mem_uIcc\n\n/-%%\n\\begin{lemma}[isIntegrable]\\label{isIntegrable}\\lean{Perron.isIntegrable}\\leanok\nLet $x>0$ and $\\sigma\\in\\R$. Then\n$$\\int_{\\R}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\nis integrable.\n\\end{lemma}\n%%-/\nlemma isIntegrable (xpos : 0 < x) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_neg_one : \u03c3 \u2260 -1) :\n    Integrable fun (t : \u211d) \u21a6 f x (\u03c3 + t * I) := by\n/-%%\n\\begin{proof}\\uses{isHolomorphicOn}\\leanok\nBy \\ref{isHolomorphicOn}, $f$ is continuous, so it is integrable on any interval.\n%%-/\n  have : Continuous (fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) := by\n    refine (isHolomorphicOn xpos).continuousOn.comp_continuous (by continuity) fun x \u21a6 not_or.mpr ?_\n    simp [Complex.ext_iff, \u03c3_ne_zero, \u03c3_ne_neg_one]\n--%% Also, $|f(x)| = \\Theta(x^{-2})$ as $x\\to\\infty$,\n  refine this.locallyIntegrable.integrable_of_isBigO_atTop_of_norm_eq_norm_neg\n    (univ_mem' fun y \u21a6 ?_) (isTheta xpos).2.isBigO \u27e8Ioi 1, Ioi_mem_atTop 1, ?_\u27e9\n--%% and $|f(-x)| = \\Theta(x^{-2})$ as $x\\to\\infty$.\n  \u00b7 show \u2016f x (\u2191\u03c3 + \u2191y * I)\u2016 = \u2016f x (\u2191\u03c3 + \u2191(-y) * I)\u2016\n    have : (\u2191\u03c3 + \u2191(-y) * I) = conj (\u2191\u03c3 + \u2191y * I) := Complex.ext (by simp) (by simp)\n    simp_rw [this, map_conj xpos.le, Complex.norm_eq_abs, abs_conj]\n--%% Since $g(x) = x^{-2}$ is integrable on $[a,\\infty)$ for any $a>0$, we conclude.\n  \u00b7 refine integrableOn_Ioi_rpow_of_lt (show (-2 : \u211d) < -1 by norm_num)\n      (show (0 : \u211d) < 1 by norm_num) |>.congr_fun (fun y hy \u21a6 ?_) measurableSet_Ioi\n    rw [rpow_neg (show (0 : \u211d) < 1 by norm_num |>.trans hy |>.le), inv_eq_one_div, rpow_two]\n--%%\\end{proof}\n\ntheorem horizontal_integral_isBigO\n    {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) (\u03bc : Measure \u211d) [IsLocallyFiniteMeasure \u03bc] :\n    (fun (y : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + y * I) \u2202\u03bc) =O[atBot \u2294 atTop]\n    fun y \u21a6 1 / y^2 := by\n  let g := fun ((\u03c3, y) : \u211d \u00d7 \u211d) \u21a6 f x (\u03c3 + y * I)\n  calc\n    _ =\u0398[atBot \u2294 atTop] fun (y : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in uIoc \u03c3' \u03c3'', g (\u03c3, y) \u2202\u03bc :=\n        isTheta_of_norm_eventuallyEq <| univ_mem'\n          fun _ \u21a6 intervalIntegral.norm_intervalIntegral_eq _ _ _ _\n    _ =O[atBot \u2294 atTop] _ :=\n      (isTheta_uniformlyOn_uIoc xpos \u03c3' \u03c3'').isBigO.set_integral_isBigO\n        measurableSet_uIoc measure_Ioc_lt_top\n\n/-%%\n\\begin{lemma}[tendsto_zero_Lower]\\label{tendsto_zero_Lower}\\lean{Perron.tendsto_zero_Lower}\\leanok\nLet $x>0$ and $\\sigma',\\sigma''\\in\\R$. Then\n$$\\int_{\\sigma'}^{\\sigma''}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\ngoes to $0$ as $t\\to-\\infty$.\n\\end{lemma}\n%%-/\nlemma tendsto_zero_Lower (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + t * I)) atBot (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nThe numerator is bounded and the denominator tends to infinity.\n\\end{proof}\n%%-/\n  have hcast : (fun (y : \u211d) \u21a6 1 / y ^ 2) =\u1da0[atBot] fun y \u21a6 (-y) ^ (-2 : \u211d) := by\n    filter_upwards [Iic_mem_atBot 0] with y hy using\n      by rw [rpow_neg (neg_nonneg.mpr hy), inv_eq_one_div, rpow_two, neg_sq]\n  exact isBigO_sup.mp (horizontal_integral_isBigO xpos \u03c3' \u03c3'' volume)\n    |>.1.trans_eventuallyEq hcast |>.trans_tendsto\n    <| tendsto_rpow_neg_atTop (by norm_num) |>.comp tendsto_neg_atBot_atTop\n\n/-%%\n\\begin{lemma}[tendsto_zero_Upper]\\label{tendsto_zero_Upper}\\lean{Perron.tendsto_zero_Upper}\\leanok\nLet $x>0$ and $\\sigma',\\sigma''\\in\\R$. Then\n$$\\int_{\\sigma'}^{\\sigma''}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\ngoes to $0$ as $t\\to\\infty$.\n\\end{lemma}\n%%-/\nlemma tendsto_zero_Upper (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + t * I)) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nThe numerator is bounded and the denominator tends to infinity.\n\\end{proof}\n%%-/\n  have hcast : (fun (y : \u211d) \u21a6 1 / y ^ 2) =\u1da0[atTop] fun y \u21a6 y ^ (-2 : \u211d) := by\n    filter_upwards [Ici_mem_atTop 0] with y hy using by rw [rpow_neg hy, inv_eq_one_div, rpow_two]\n  refine isBigO_sup.mp (horizontal_integral_isBigO xpos \u03c3' \u03c3'' volume)\n    |>.2.trans_eventuallyEq hcast |>.trans_tendsto <| tendsto_rpow_neg_atTop (by norm_num)\n\nlemma contourPull {\u03c3' \u03c3'' : \u211d} (xpos : 0 < x) (h\u03c30 : 0 \u2209 [[\u03c3', \u03c3'']]) (h\u03c31 : -1 \u2209 [[\u03c3', \u03c3'']]) :\n    VerticalIntegral (f x) \u03c3' = VerticalIntegral (f x) \u03c3'' := by\n  refine verticalIntegral_eq_verticalIntegral ((isHolomorphicOn xpos).mono ?_)\n    (tendsto_zero_Lower xpos \u03c3' \u03c3'') (tendsto_zero_Upper xpos \u03c3' \u03c3'')\n    (isIntegrable xpos (fun h \u21a6 h\u03c30 (h \u25b8 left_mem_uIcc)) (fun h \u21a6 h\u03c31 (h \u25b8 left_mem_uIcc)))\n    (isIntegrable xpos (fun h \u21a6 h\u03c30 (h \u25b8 right_mem_uIcc)) (fun h \u21a6 h\u03c31 (h \u25b8 right_mem_uIcc)))\n  rintro \u27e8x, y\u27e9 \u27e8hx, hy\u27e9 \u27e8hc | hc\u27e9 <;> simp_all [Complex.ext_iff]\n\n/-%%\nWe are ready for the first case of the Perron formula, namely when $x<1$:\n\\begin{lemma}[formulaLtOne]\\label{formulaLtOne}\\lean{Perron.formulaLtOne}\\leanok\nFor $x>0$, $\\sigma>0$, and $x<1$, we have\n$$\n\\frac1{2\\pi i}\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds =0.\n$$\n\\end{lemma}\n%%-/\nlemma formulaLtOne (xpos : 0 < x) (x_lt_one : x < 1) (\u03c3_pos : 0 < \u03c3)\n    : VerticalIntegral (f x) \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{isHolomorphicOn, HolomorphicOn.vanishesOnRectangle, integralPosAux,\nvertIntBound, limitOfConstant,\ntendsto_rpow_atTop_nhds_zero_of_norm_lt_one,\ntendsto_zero_Lower, tendsto_zero_Upper, isIntegrable}\n  Let $f(s) = x^s/(s(s+1))$. Then $f$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n  The rectangle integral of $f$ with corners $\\sigma-iT$ and $\\sigma+iT$ is zero.\n  The limit of this rectangle integral as $T\\to\\infty$ is $\\int_{(\\sigma')}-\\int_{(\\sigma)}$.\n  Therefore, $\\int_{(\\sigma')}=\\int_{(\\sigma)}$.\n%%-/\n  have h_contourPull (\u03c3' \u03c3'' : \u211d) (\u03c3'pos : 0 < \u03c3') (\u03c3''pos : 0 < \u03c3'') :\n      VerticalIntegral (f x) \u03c3' = VerticalIntegral (f x) \u03c3'' :=\n    contourPull xpos (not_mem_uIcc_of_lt \u03c3'pos \u03c3''pos)\n      (not_mem_uIcc_of_lt (by linarith) (by linarith))\n--%% But we also have the bound $\\int_{(\\sigma')} \\leq x^{\\sigma'} * C$, where\n--%% $C=\\int_\\R\\frac{1}{|(1+t)(1+t+1)|}dt$.\n  have VertIntBound : \u2203 C > 0, \u2200 \u03c3' > 1, \u2016VerticalIntegral (f x) \u03c3'\u2016 \u2264 x^\u03c3' * C := by\n    let C := \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt)\n    exact \u27e8C, integralPosAux, fun _ \u21a6 vertIntBound xpos\u27e9\n--%% Therefore $\\int_{(\\sigma')}\\to 0$ as $\\sigma'\\to\\infty$.\n  have AbsVertIntTendsto : Tendsto (Complex.abs \u2218 (VerticalIntegral (f x))) atTop (\ud835\udcdd 0) := by\n    obtain \u27e8C, _, hC\u27e9 := VertIntBound\n    have := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one xpos x_lt_one C\n    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds this\n    \u00b7 filter_upwards; exact fun _ \u21a6 Complex.abs.nonneg' _\n    \u00b7 filter_upwards [eventually_gt_atTop 1]; exact hC\n  have VertIntTendsto : Tendsto (VerticalIntegral (f x)) atTop (\ud835\udcdd 0) :=\n    tendsto_zero_iff_norm_tendsto_zero.mpr AbsVertIntTendsto\n  --%% So pulling contours gives $\\int_{(\\sigma)}=0$.\n  exact limitOfConstant \u03c3_pos h_contourPull VertIntTendsto\n--%%\\end{proof}\n\n/-%%\nThe second case is when $x>1$.\nHere are some auxiliary lemmata for the second case.\nTODO: Move to more general section\n%%-/\n\ntheorem HolomorphicOn.upperUIntegral_eq_zero {f : \u2102 \u2192 \u2102} {\u03c3 \u03c3' T : \u211d} (h\u03c3 : \u03c3 \u2264 \u03c3')\n    (hf : HolomorphicOn f {z : \u2102 | \u03c3 \u2264 z.re \u2227 z.re \u2264 \u03c3' \u2227 T \u2264 z.im})\n    (htop : Tendsto (fun y : \u211d \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x + \u2191y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3 + \u2191y * I))\n    (hright : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3' + \u2191y * I)) :\n    UpperUIntegral f \u03c3 \u03c3' T = 0 := by\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_UpperU htop hleft hright)\n  apply EventuallyEq.tendsto\n  filter_upwards [eventually_ge_atTop T]\n  refine fun _ hTU \u21a6 hf.vanishesOnRectangle fun _ \u21a6 ?_\n  rw [mem_Rect (by simp [h\u03c3]) (by simp [hTU])]\n  simpa using by tauto\n\ntheorem HolomorphicOn.lowerUIntegral_eq_zero {f : \u2102 \u2192 \u2102} {\u03c3 \u03c3' T : \u211d} (h\u03c3 : \u03c3 \u2264 \u03c3')\n    (hf : HolomorphicOn f {z : \u2102 | \u03c3 \u2264 z.re \u2227 z.re \u2264 \u03c3' \u2227 z.im \u2264 -T})\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3 + \u2191y * I))\n    (hright : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3' + \u2191y * I)) :\n    LowerUIntegral f \u03c3 \u03c3' T = 0 := by\n  suffices h : - LowerUIntegral f \u03c3 \u03c3' T = 0 by exact neg_eq_zero.mp h\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_LowerU hbot hleft hright)\n  apply EventuallyEq.tendsto\n  filter_upwards [eventually_ge_atTop T]\n  refine fun _ hTU \u21a6 hf.vanishesOnRectangle fun _ \u21a6 ?_\n  rw [mem_Rect (by simp [h\u03c3]) (by simp [hTU])]\n  simpa using by tauto\n\nlemma sPlusOneNeZero {s : \u2102} (s_ne_neg_one : s \u2260 -1) : s + 1 \u2260 0 :=\n  fun h \u21a6 s_ne_neg_one (add_eq_zero_iff_eq_neg.mp h)\n\n/-%%\n\\begin{lemma}[keyIdentity]\\label{keyIdentity}\\lean{Perron.keyIdentity}\\leanok\nLet $x\\in \\R$ and $s \\ne 0, -1$. Then\n$$\n\\frac{x^\\sigma}{s(1+s)} = \\frac{x^\\sigma}{s} - \\frac{x^\\sigma}{1+s}\n$$\n\\end{lemma}\n%%-/\nlemma keyIdentity (x : \u211d) {s : \u2102} (s_ne_zero : s \u2260 0) (s_ne_neg_one : s \u2260 -1) :\n    (x : \u2102) ^ s / (s * (s + 1))\n      = (x : \u2102) ^ s / s - (x : \u2102) ^ s / (s + 1) := by\n    field_simp [sPlusOneNeZero, mul_ne_zero]; ring_nf\n/-%%\n\\begin{proof}\\leanok\nBy ring.\n\\end{proof}\n%%-/\n\nvariable  {\u03b1 \u03b2 : Type*} [LinearOrder \u03b2] [NoMaxOrder \u03b2] [TopologicalSpace \u03b2] [ClosedIciTopology \u03b2]\n  {y : \u03b2} {l : Filter \u03b1}\n\nlemma _root_.Filter.Tendsto.eventually_bddAbove {f : \u03b1 \u2192 \u03b2} (hf : Tendsto f l (\ud835\udcdd y)) :\n    \u2200\u1da0 s in l.smallSets, BddAbove (f '' s) := by\n  obtain \u27e8y', hy'\u27e9 := exists_gt y\n  obtain \u27e8s, hsl, hs\u27e9 := (eventually_le_of_tendsto_lt hy' hf).exists_mem\n  simp_rw [Filter.eventually_smallSets, bddAbove_def]\n  refine \u27e8s, hsl, fun t ht \u21a6 \u27e8y', fun y hy \u21a6 ?_\u27e9\u27e9\n  obtain \u27e8x, hxt, hxy\u27e9 := hy\n  exact hxy \u25b8 hs x (ht hxt)\n\nlemma bddAbove_square_of_tendsto {f : \u2102 \u2192 \u03b2} {x : \u2102} (hf : Tendsto f (\ud835\udcdd[\u2260] x) (\ud835\udcdd y)) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, BddAbove (f '' (Square x c \\ {x})) := by\n  obtain \u27e8t, htf, ht\u27e9 := eventually_smallSets.mp hf.eventually_bddAbove\n  obtain \u27e8\u03b5, h\u03b50, h\u03b5\u27e9 := nhdsWithin_hasBasis (nhds_hasBasis_square x) {x}\u1d9c |>.1 t |>.mp htf\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' h\u03b50] with \u03b5' \u27e8h\u03b5'0, h\u03b5'\u27e9\n  exact ht _ <| (diff_subset_diff (square_subset_square h\u03b5'0 h\u03b5'.le) subset_rfl).trans h\u03b5\n\n/-%%\n\\begin{lemma}[diffBddAtZero]\\label{diffBddAtZero}\\lean{Perron.diffBddAtZero}\\leanok\nLet $x>0$. Then for $0 < c < 1 /2$, we have that the function\n$$\ns \u21a6 \\frac{x^s}{s(s+1)} - \\frac1s\n$$\nis bounded above on the rectangle with corners at $-c-i*c$ and $c+i*c$ (except at $s=0$).\n\\end{lemma}\n%%-/\nlemma diffBddAtZero {x : \u211d} (xpos : 0 < x) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, BddAbove ((norm \u2218 (fun (s : \u2102) \u21a6 (x : \u2102) ^ s / (s * (s + 1)) - 1 / s)) ''\n    (Square 0 c \\ {0})) := by\n/-%%\n\\begin{proof}\\uses{keyIdentity}\\leanok\nApplying Lemma \\ref{keyIdentity}, the\n function $s \u21a6 x^s/s(s+1) - 1/s = x^s/s - x^0/s - x^s/(1+s)$. The last term is bounded for $s$\n away from $-1$. The first two terms are the difference quotient of the function $s \u21a6 x^s$ at\n $0$; since it's differentiable, the difference remains bounded as $s\\to 0$.\n\\end{proof}\n%%-/\n  apply bddAbove_square_of_tendsto\n  suffices Tendsto (norm \u2218 (fun (s : \u2102) \u21a6 \u2191x ^ s / s - \u2191x ^ (0 : \u2102) / s - \u2191x ^ s / (1 + s)))\n      (\ud835\udcdd[\u2260] 0) (\ud835\udcdd (\u2016(deriv (fun (s : \u2102) \u21a6 (x : \u2102) ^ s) 0) - x ^ (0 : \u2102) / (1 + 0)\u2016)) by\n    apply this.congr'\n    filter_upwards [diff_mem_nhdsWithin_compl (isOpen_compl_singleton.mem_nhds\n      (Set.mem_compl_singleton_iff.mpr (by norm_num : (0 : \u2102) \u2260 -1))) {0}] with s hs\n    rw [Function.comp_apply, Function.comp_apply, keyIdentity _ hs.2 hs.1, cpow_zero]; ring_nf\n  have hx0 : (x : \u2102) \u2260 0 := slitPlane_ne_zero (.inl xpos)\n  refine (Tendsto.sub ?_ (tendsto_nhdsWithin_of_tendsto_nhds ?_)).norm\n  \u00b7 convert hasDerivAt_iff_tendsto_slope.mp\n      (differentiableAt_id'.const_cpow (.inl hx0)).hasDerivAt using 2\n    rw [slope_def_field]; ring\n  \u00b7 exact (continuous_id.const_cpow (.inl hx0)).tendsto 0\n      |>.div (tendsto_const_nhds.add tendsto_id) (by norm_num)\n\n/-%%\n\\begin{lemma}[diffBddAtNegOne]\\label{diffBddAtNegOne}\\lean{Perron.diffBddAtNegOne}\\leanok\nLet $x>0$. Then for $0 < c < 1 /2$, we have that the function\n$$\ns \u21a6 \\frac{x^s}{s(s+1)} - \\frac{-x^{-1}}{s+1}\n$$\nis bounded above on the rectangle with corners at $-1-c-i*c$ and $-1+c+i*c$ (except at $s=-1$).\n\\end{lemma}\n%%-/\nlemma diffBddAtNegOne {x : \u211d} (xpos : 0 < x) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0,\n    BddAbove ((norm \u2218 (fun (s : \u2102) \u21a6 (x : \u2102) ^ s / (s * (s + 1)) - (-x\u207b\u00b9) / (s+1))) ''\n      (Square (-1) c \\ {-1})) := by\n/-%%\n\\begin{proof}\\uses{keyIdentity}\\leanok\nApplying Lemma \\ref{keyIdentity}, the\n function $s \u21a6 x^s/s(s+1) - x^{-1}/(s+1) = x^s/s - x^s/(s+1) - (-x^{-1})/(s+1)$. The first term is bounded for $s$\n away from $0$. The last two terms are the difference quotient of the function $s \u21a6 x^s$ at\n $-1$; since it's differentiable, the difference remains bounded as $s\\to -1$.\n\\end{proof}\n%%-/\n  apply bddAbove_square_of_tendsto\n  suffices Tendsto (norm \u2218 (fun (s : \u2102) \u21a6 \u2191x ^ s / s - (\u2191x ^ s / (s + 1) - x\u207b\u00b9 / (s + 1))))\n      (\ud835\udcdd[\u2260] (-1)) (\ud835\udcdd (\u2016x ^ (-1 : \u2102) / -1 - (deriv (fun (s : \u2102) \u21a6 (x : \u2102) ^ s) (-1))\u2016)) by\n    apply this.congr'\n    filter_upwards [diff_mem_nhdsWithin_compl (isOpen_compl_singleton.mem_nhds\n      (Set.mem_compl_singleton_iff.mpr (by norm_num : (-1 : \u2102) \u2260 0))) {-1}] with s hs\n    rw [Function.comp_apply, Function.comp_apply, keyIdentity _ hs.1 hs.2]\n    ring_nf\n  have hx0 : (x : \u2102) \u2260 0 := slitPlane_ne_zero (.inl xpos)\n  refine (Tendsto.sub (tendsto_nhdsWithin_of_tendsto_nhds ?_) ?_).norm\n  \u00b7 exact ((continuous_id.const_cpow (.inl hx0)).tendsto _).div tendsto_id (by norm_num)\n  \u00b7 convert hasDerivAt_iff_tendsto_slope.mp\n      (differentiableAt_id'.const_cpow (.inl hx0)).hasDerivAt using 2\n    rw [slope_def_field, cpow_neg_one, ofReal_inv]; ring\n\n/-%%\n\\begin{lemma}[residueAtZero]\\label{residueAtZero}\\lean{Perron.residueAtZero}\\leanok\nLet $x>0$. Then for all sufficiently small $c>0$, we have that\n$$\n\\frac1{2\\pi i}\n\\int_{-c-i*c}^{c+ i*c}\\frac{x^s}{s(s+1)}ds = 1.\n$$\n\\end{lemma}\n%%-/\nlemma residueAtZero (xpos : 0 < x) : \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0,\n    RectangleIntegral' (f x) (-c - c * I) (c + c * I) = 1 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{diffBddAtZero, ResidueTheoremOnRectangleWithSimplePole,\nexistsDifferentiableOn_of_bddAbove}\nFor $c>0$ sufficiently small,\n%%-/\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < 1 / 2), diffBddAtZero xpos]\n  intro c hc bddAbove\n  obtain \u27e8cpos, _\u27e9 := hc\n  have RectSub : Square 0 c \\ {0} \u2286 {0, -1}\u1d9c := by\n    refine fun s \u27e8hs, hs0\u27e9 \u21a6 not_or.mpr \u27e8hs0, ?_\u27e9\n    rw [Square, mem_Rect (by simpa using by linarith) (by simp [cpos.le])] at hs\n    replace hs : -c \u2264 s.re \u2227 s.re \u2264 c \u2227 -c \u2264 s.im \u2227 s.im \u2264 c := by simpa using hs\n    simpa [Complex.ext_iff] using fun h \u21a6 by linarith\n  have fHolo : HolomorphicOn (f x) (Square 0 c \\ {0}) := (isHolomorphicOn xpos).mono RectSub\n  have f1Holo : HolomorphicOn ((f x) - (fun (s : \u2102) \u21a6 1 / s)) (Square 0 c \\ {0}) :=\n    fHolo.sub (by simpa using differentiableOn_inv.mono fun s hs \u21a6 hs.2)\n\n  have RectMemNhds : Square 0 c \u2208 \ud835\udcdd 0 := square_mem_nhds 0 (ne_of_gt cpos)\n/-%% $x^s/(s(s+1))$ is equal to $1/s$ plus a function, $g$, say,\nholomorphic in the whole rectangle (by Lemma \\ref{diffBddAtZero}).\n%%-/\n  obtain \u27e8g, gHolo, g_eq_fDiff\u27e9 := existsDifferentiableOn_of_bddAbove RectMemNhds f1Holo bddAbove\n  simp_rw [Square, add_zero] at fHolo gHolo RectMemNhds\n\n--%% Now apply Lemma \\ref{ResidueTheoremOnRectangleWithSimplePole}.\n  refine ResidueTheoremOnRectangleWithSimplePole ?_ ?_ RectMemNhds gHolo ?_\n  any_goals simpa using cpos.le\n  convert g_eq_fDiff using 3 <;> simp [Square]\n--%%\\end{proof}\n\nlemma residueAtNegOne (xpos : 0 < x) : \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0,\n    RectangleIntegral' (f x) (-c - c * I - 1) (c + c * I - 1) = -x\u207b\u00b9 := by\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < 1 / 2), diffBddAtNegOne xpos]\n  intro c hc bddAbove\n  obtain \u27e8cpos, _\u27e9 := hc\n  have h_mem {s : \u2102} (hs : s \u2208 Square (-1) c) :\n      -c \u2264 s.re + 1 \u2227 s.re + 1 \u2264 c \u2227 -c \u2264 s.im \u2227 s.im \u2264 c := by\n    rw [Square, mem_Rect (by simpa using by linarith) (by simp [cpos.le])] at hs\n    simpa using hs\n  have RectSub : Square (-1) c \\ {-1} \u2286 {0, -1}\u1d9c := by\n    refine fun s \u27e8hs, hs1\u27e9 \u21a6 not_or.mpr \u27e8?_, hs1\u27e9\n    simpa [Complex.ext_iff] using fun _ _ \u21a6 by linarith [h_mem hs]\n  have fHolo : HolomorphicOn (f x) (Square (-1) c \\ {-1}) := (isHolomorphicOn xpos).mono RectSub\n  have f1Holo : HolomorphicOn ((f x) - (fun (s : \u2102) \u21a6 -x\u207b\u00b9 / (s + 1))) (Square (-1) c \\ {-1}) := by\n    refine fHolo.sub <| (differentiableOn_const _).neg.div ?_ fun x hx \u21a6 sPlusOneNeZero hx.2\n    exact differentiableOn_id.add (differentiableOn_const 1)\n  have RectMemNhds : Square (-1) c \u2208 \ud835\udcdd (-1) := square_mem_nhds (-1) (ne_of_gt cpos)\n  obtain \u27e8g, gHolo, g_eq_fDiff\u27e9 := existsDifferentiableOn_of_bddAbove RectMemNhds f1Holo bddAbove\n  simp_rw [Square] at fHolo gHolo RectMemNhds\n  refine ResidueTheoremOnRectangleWithSimplePole ?_ ?_ RectMemNhds gHolo ?_\n  \u00b7 simpa using cpos.le\n  \u00b7 simpa using cpos.le\n  \u00b7 convert g_eq_fDiff using 3; simp\n\n/-%%\n\\begin{lemma}[residuePull1]\\label{residuePull1}\\lean{Perron.residuePull1}\\leanok\nFor $x>1$ (of course $x>0$ would suffice) and $\\sigma>0$, we have\n$$\n\\frac1{2\\pi i}\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds =1\n+\n\\frac 1{2\\pi i}\n\\int_{(-1/2)}\\frac{x^s}{s(s+1)}ds.\n$$\n\\end{lemma}\n%%-/\n", "theoremStatement": "lemma residuePull1 (x_gt_one : 1 < x) (\u03c3_pos : 0 < \u03c3) :\n    VerticalIntegral' (f x) \u03c3 = 1 + VerticalIntegral' (f x) (-1 / 2) ", "theoremName": "Perron.residuePull1", "fileCreated": {"commit": "70815e04f04e213b27a1f0756724b06c69b56da1", "date": "2024-02-06"}, "theoremCreated": {"commit": "42a6c5d846f0077dca17a99c2b5d587defdbe155", "date": "2024-02-06"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/PerronFormula.lean", "module": "PrimeNumberTheoremAnd.PerronFormula", "jsonFile": "PrimeNumberTheoremAnd.PerronFormula.jsonl", "positionMetadata": {"lineInFile": 865, "tokenPositionInFile": 41068, "theoremPositionInFile": 43}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 362, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", 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"Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Integral.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n/-%%\n\\begin{proof}\\leanok\n\\uses{residueAtZero}\nWe pull to a square with corners at $-c-i*c$ and $c+i*c$ for $c>0$\nsufficiently small.\nBy Lemma \\ref{residueAtZero}, the integral over this square is equal to $1$.\n\\end{proof}\n%%-/\n  apply eq_add_of_sub_eq\n  have xpos : 0 < x := zero_lt_one.trans x_gt_one\n  have hf : HolomorphicOn (f x) (Icc (-1 / 2) \u03c3 \u00d7\u2102 univ \\ {0}) :=\n    (isHolomorphicOn xpos).mono fun s \u27e8\u27e8\u27e8_, _\u27e9, _\u27e9, hs0\u27e9 hc \u21a6 hc.casesOn\n      (fun hc \u21a6 hs0 hc) (fun hc \u21a6 by linarith [show s.re = -1 from congrArg _ hc])\n  have := (residueAtZero xpos).and <| verticalIntegral_sub_verticalIntegral_eq_squareIntegral\n    (by simpa using \u27e8by linarith, by linarith\u27e9) hf\n    (tendsto_zero_Lower xpos _ _) (tendsto_zero_Upper xpos _ _)\n    (isIntegrable xpos (by norm_num) (by norm_num)) (isIntegrable xpos (by linarith) (by linarith))\n  obtain \u27e8c, hcf, hc\u27e9 := this.exists_mem\n  obtain \u27e8\u03b5, h\u03b5, h\u03b5c\u27e9 := Metric.mem_nhdsWithin_iff.mp hcf\n  obtain h\u03b5 := hc (\u03b5/2) (h\u03b5c \u27e8mem_ball_iff_norm.mpr (by simp [abs_of_pos h\u03b5, h\u03b5]), half_pos h\u03b5\u27e9)\n  rw [VerticalIntegral', \u2190 smul_sub, h\u03b5.2, \u2190 RectangleIntegral', add_zero, add_zero, h\u03b5.1]", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1126}}
+{"srcContext": "/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Kontorovich, David Loeffler, Heather Macbeth\n-/\nimport Mathlib.Analysis.Calculus.ParametricIntegral\nimport Mathlib.Analysis.Fourier.AddCircle\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Calculus.FDeriv.Analytic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space\n\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\n\n/-!\n# Derivatives of the Fourier transform\n\nIn this file we compute the Fr\u00e9chet derivative of the Fourier transform of `f`, where `f` is a\nfunction such that both `f` and `v \u21a6 \u2016v\u2016 * \u2016f v\u2016` are integrable. Here the Fourier transform is\nunderstood as an operator `(V \u2192 E) \u2192 (W \u2192 E)`, where `V` and `W` are normed `\u211d`-vector spaces\nand the Fourier transform is taken with respect to a continuous `\u211d`-bilinear\npairing `L : V \u00d7 W \u2192 \u211d` and a given reference measure `\u03bc`.\n\nWe also investigate higher derivatives: Assuming that `\u2016v\u2016^n * \u2016f v\u2016` is integrable, we show\nthat the Fourier transform of `f` is `C^n`.\n\nWe also study in a parallel way the Fourier transform of the derivative, which is obtained by\ntensoring the Fourier transform of the original function with the bilinear form.\n\nWe give specialized versions of these results on inner product spaces (where `L` is the scalar\nproduct) and on the real line, where we express the one-dimensional derivative in more concrete\nterms, as the Fourier transform of `-2\u03c0I x * f x` (or `(-2\u03c0I x)^n * f x` for higher derivatives).\n\n## Main definitions and results\n\nWe introduce two convenience definitions:\n\n* `VectorFourier.fourierSMulRight L f`: given `f : V \u2192 E` and `L` a bilinear pairing\n  between `V` and `W`, then this is the function `fun v \u21a6 -(2 * \u03c0 * I) (L v \u2b1d) \u2022 f v`,\n  from `V` to `Hom (W, E)`.\n  This is essentially `ContinousLinearMap.smulRight`, up to the factor `- 2\u03c0I` designed to make sure\n  that the Fourier integral of `fourierSMulRight L f` is the derivative of the Fourier\n  integral of `f`.\n* `VectorFourier.fourierPowSMulRight` is the higher order analogue for higher derivatives:\n  `fourierPowSMulRight L f v n` is informally `(-(2 * \u03c0 * I))^n (L v \u2b1d)^n \u2022 f v`, in\n  the space of continuous multilinear maps `W [\u00d7n]\u2192L[\u211d] E`.\n\nWith these definitions, the statements read as follows, first in a general context\n(arbitrary `L` and `\u03bc`):\n\n* `VectorFourier.hasFDerivAt_fourierIntegral`: the Fourier integral of `f` is differentiable, with\n    derivative the Fourier integral of `fourierSMulRight L f`.\n* `VectorFourier.differentiable_fourierIntegral`: the Fourier integral of `f` is differentiable.\n* `VectorFourier.fderiv_fourierIntegral`: formula for the derivative of the Fourier integral of `f`.\n* `VectorFourier.fourierIntegral_fderiv`: formula for the Fourier integral of the derivative of `f`.\n* `VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` has an explicit Taylor series up to order `N`, given by the Fourier\n  integrals of `fun v \u21a6 fourierPowSMulRight L f v n`.\n* `VectorFourier.contDiff_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` is `C^n`.\n* `VectorFourier.iteratedFDeriv_fourierIntegral`: under suitable integrability conditions,\n  explicit formula for the `n`-th derivative of the Fourier integral of `f`, as the Fourier\n  integral of `fun v \u21a6 fourierPowSMulRight L f v n`.\n\nThese statements are then specialized to the case of the usual Fourier transform on\nfinite-dimensional inner product spaces with their canonical Lebesgue measure (covering in\nparticular the case of the real line), replacing the namespace `VectorFourier` by\nthe namespace `Real` in the above statements.\n\nWe also give specialized versions of the one-dimensional real derivative (and iterated derivative)\nin `Real.deriv_fourierIntegral` and `Real.iteratedDeriv_fourierIntegral`.\n-/\n\nnoncomputable section\n\nopen Real Complex MeasureTheory Filter TopologicalSpace\n\nopen scoped FourierTransform Topology BigOperators\n\n-- without this local instance, Lean tries first the instance\n-- `secondCountableTopologyEither_of_right` (whose priority is 100) and takes a very long time to\n-- fail. Since we only use the left instance in this file, we make sure it is tried first.\nattribute [local instance 101] secondCountableTopologyEither_of_left\n\nnamespace Real\n\nlemma differentiable_fourierChar : Differentiable \u211d (\ud835\udc1e \u00b7 : \u211d \u2192 \u2102) :=\n  fun x \u21a6 (Real.hasDerivAt_fourierChar x).differentiableAt\n\nlemma deriv_fourierChar (x : \u211d) : deriv (\ud835\udc1e \u00b7 : \u211d \u2192 \u2102) x = 2 * \u03c0 * I * \ud835\udc1e x :=\n  (Real.hasDerivAt_fourierChar x).deriv\n\nvariable {V W : Type*} [NormedAddCommGroup V] [NormedSpace \u211d V]\n  [NormedAddCommGroup W] [NormedSpace \u211d W] (L : V \u2192L[\u211d] W \u2192L[\u211d] \u211d)\n\nlemma hasFDerivAt_fourierChar_neg_bilinear_right (v : V) (w : W) :\n    HasFDerivAt (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102))\n      ((-2 * \u03c0 * I * \ud835\udc1e (-L v w)) \u2022 (ofRealCLM \u2218L (L v))) w := by\n  have ha : HasFDerivAt (fun w' : W \u21a6 L v w') (L v) w := ContinuousLinearMap.hasFDerivAt (L v)\n  convert (hasDerivAt_fourierChar (-L v w)).hasFDerivAt.comp w ha.neg\n  ext y\n  simp only [neg_mul, ContinuousLinearMap.coe_smul', ContinuousLinearMap.coe_comp', Pi.smul_apply,\n    Function.comp_apply, ofRealCLM_apply, smul_eq_mul, ContinuousLinearMap.comp_neg,\n    ContinuousLinearMap.neg_apply, ContinuousLinearMap.smulRight_apply,\n    ContinuousLinearMap.one_apply, real_smul, neg_inj]\n  ring\n\nlemma fderiv_fourierChar_neg_bilinear_right_apply (v : V) (w y : W) :\n    fderiv \u211d (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102)) w y = -2 * \u03c0 * I * L v y * \ud835\udc1e (-L v w) := by\n  simp [(hasFDerivAt_fourierChar_neg_bilinear_right L v w).fderiv]\n  ring\n\nlemma differentiable_fourierChar_neg_bilinear_right (v : V) :\n    Differentiable \u211d (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102)) :=\n  fun w \u21a6 (hasFDerivAt_fourierChar_neg_bilinear_right L v w).differentiableAt\n\nlemma hasFDerivAt_fourierChar_neg_bilinear_left (v : V) (w : W) :\n    HasFDerivAt (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102))\n      ((-2 * \u03c0 * I * \ud835\udc1e (-L v w)) \u2022 (ofRealCLM \u2218L (L.flip w))) v :=\n  hasFDerivAt_fourierChar_neg_bilinear_right L.flip w v\n\nlemma fderiv_fourierChar_neg_bilinear_left_apply (v y : V) (w : W) :\n    fderiv \u211d (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102)) v y = -2 * \u03c0 * I * L y w * \ud835\udc1e (-L v w) := by\n  simp [(hasFDerivAt_fourierChar_neg_bilinear_left L v w).fderiv]\n  ring\n\nlemma differentiable_fourierChar_neg_bilinear_left (w : W) :\n    Differentiable \u211d (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102)) :=\n  fun v \u21a6 (hasFDerivAt_fourierChar_neg_bilinear_left L v w).differentiableAt\n\nend Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E]\n\nnamespace VectorFourier\n\nvariable {V W : Type*} [NormedAddCommGroup V] [NormedSpace \u211d V]\n  [NormedAddCommGroup W] [NormedSpace \u211d W] (L : V \u2192L[\u211d] W \u2192L[\u211d] \u211d) (f : V \u2192 E)\n\nvariable {f}\n\n/-- The Fourier integral of the derivative of a function is obtained by multiplying the Fourier\nintegral of the original function by `-L w v`. -/\ntheorem fourierIntegral_fderiv [MeasurableSpace V] [BorelSpace V] [FiniteDimensional \u211d V]\n    {\u03bc : Measure V} [Measure.IsAddHaarMeasure \u03bc]\n    (hf : Integrable f \u03bc) (h'f : Differentiable \u211d f) (hf' : Integrable (fderiv \u211d f) \u03bc) :\n    fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 (fderiv \u211d f)\n      = fourierSMulRight (-L.flip) (fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 f) := by\n  ext w y\n  let g : V \u2192 \u2102 := fun v \u21a6 \ud835\udc1e (-L v w)\n  have J : Integrable (fun v \u21a6 \ud835\udc1e (-(L v) w) \u2022 fderiv \u211d f v) \u03bc :=\n    (fourierIntegral_convergent_iff' _ _).2 hf'\n  /- First rewrite things in a simplified form, without any real change. -/\n  suffices \u222b x, g x \u2022 fderiv \u211d f x y \u2202\u03bc = \u222b x, (2 * \u2191\u03c0 * I * L y w * g x) \u2022 f x \u2202\u03bc by\n    simpa only [fourierIntegral, ContinuousLinearMap.toLinearMap\u2082_apply,\n      ContinuousLinearMap.integral_apply J, ContinuousLinearMap.coe_smul', Pi.smul_apply,\n      fourierSMulRight_apply, ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply, \u2190\n      integral_smul, neg_smul, smul_neg, \u2190 smul_smul, Complex.coe_smul, neg_neg]\n  have A x : fderiv \u211d g x y = - 2 * \u2191\u03c0 * I * L y w * g x :=\n    fderiv_fourierChar_neg_bilinear_left_apply _ _ _ _\n  /- Key step: integrate by parts with respect to `y` to switch the derivative from `f` to `g`. -/\n  rw [integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable, \u2190 integral_neg]\n  \u00b7 congr with x\n    simp only [A, neg_mul, neg_smul, neg_neg]\n  \u00b7 have : Integrable (fun x \u21a6 (-(2 * \u2191\u03c0 * I * \u2191((L y) w)) \u2022 ((g x : \u2102) \u2022 f x))) \u03bc :=\n      ((fourierIntegral_convergent_iff' _ _).2 hf).smul _\n    convert this using 2 with x\n    simp [A, smul_smul]\n  \u00b7 exact (fourierIntegral_convergent_iff' _ _).2 (hf'.apply_continuousLinearMap _)\n  \u00b7 exact (fourierIntegral_convergent_iff' _ _).2 hf\n  \u00b7 exact differentiable_fourierChar_neg_bilinear_left _ _\n  \u00b7 exact h'f\n\nopen ContinuousMultilinearMap\n\nvariable [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {\u03bc : Measure V}\n\n/-- The Fourier integral of the `n`-th derivative of a function is obtained by multiplying the\nFourier integral of the original function by `(2\u03c0I L w \u2b1d )^n`. -/\ntheorem fourierIntegral_iteratedFDeriv [FiniteDimensional \u211d V]\n    {\u03bc : Measure V} [Measure.IsAddHaarMeasure \u03bc] {N : \u2115\u221e} (hf : ContDiff \u211d N f)\n    (h'f : \u2200 (n : \u2115), n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f) \u03bc) {n : \u2115} (hn : n \u2264 N) :\n    fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 (iteratedFDeriv \u211d n f)\n      = (fun w \u21a6 fourierPowSMulRight (-L.flip) (fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 f) w n) := by\n  induction n with\n  | zero =>\n    ext w m\n    have I : Integrable (fun v \u21a6 \ud835\udc1e (- L v w) \u2022 iteratedFDeriv \u211d 0 f v) \u03bc :=\n      (fourierIntegral_convergent_iff' _ _).2 (h'f 0 bot_le)\n    simp only [Nat.zero_eq, fourierIntegral, ContinuousLinearMap.toLinearMap\u2082_apply,\n      integral_apply I, smul_apply, iteratedFDeriv_zero_apply, fourierPowSMulRight_apply, pow_zero,\n      Finset.univ_eq_empty, ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply,\n      Finset.prod_empty, one_smul]\n  | succ n ih =>\n    ext w m\n    -- instance on next line should not be necessary, but proof breaks down without it.\n    let NS : NormedSpace \u211d (V [\u00d7n]\u2192L[\u211d] E) := by infer_instance\n    have J : Integrable (fderiv \u211d (iteratedFDeriv \u211d n f)) \u03bc := by\n      specialize h'f (n + 1) hn\n      simp_rw [iteratedFDeriv_succ_eq_comp_left] at h'f\n      exact (LinearIsometryEquiv.integrable_comp_iff _).1 h'f\n    suffices H : (fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 (fderiv \u211d (iteratedFDeriv \u211d n f)) w)\n          (m 0) (Fin.tail m) =\n        (-(2 * \u03c0 * I)) ^ (n + 1) \u2022 (\u220f x : Fin (n + 1), -L (m x) w) \u2022 \u222b v, \ud835\udc1e (-L v w) \u2022 f v \u2202\u03bc by\n      have A : \u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v (m 0)) (Fin.tail m) \u2202\u03bc\n          = (\u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v (m 0)) \u2202\u03bc) (Fin.tail m) := by\n        rw [integral_apply]\n        \u00b7 simp only [smul_apply]\n        \u00b7 exact (fourierIntegral_convergent_iff' L w).2 (J.apply_continuousLinearMap _)\n      have B : \u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v (m 0)) \u2202\u03bc =\n          (\u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v) \u2202\u03bc) (m 0) := by\n        rw [ContinuousLinearMap.integral_apply]\n        \u00b7 simp only [ContinuousLinearMap.coe_smul', Pi.smul_apply]\n        \u00b7 exact (fourierIntegral_convergent_iff' L w).2 J\n      simp only [fourierIntegral, ContinuousLinearMap.toLinearMap\u2082_apply,\n        integral_apply ((fourierIntegral_convergent_iff' L w).2 (h'f _ hn)), smul_apply,\n        iteratedFDeriv_succ_apply_left, fourierPowSMulRight_apply, ContinuousLinearMap.neg_apply,\n        ContinuousLinearMap.flip_apply, A, B]\n      exact H\n    have h'n : n < N := lt_of_lt_of_le (by simp [-Nat.cast_succ]) hn\n    rw [fourierIntegral_fderiv]\n    \u00b7 have A : \u2200 (x : \u211d) (v : E), x \u2022 v = (x : \u2102) \u2022 v := fun x v \u21a6 rfl\n      simp only [ih h'n.le, fourierSMulRight_apply, ContinuousLinearMap.neg_apply,\n        ContinuousLinearMap.flip_apply, neg_smul, smul_neg, neg_neg, smul_apply,\n        fourierPowSMulRight_apply, A, smul_smul]\n      congr 1\n      have B : \u2200 (i : Fin n), Fin.tail m i = m (Fin.succ i) := fun i \u21a6 rfl\n      simp only [ofReal_prod, ofReal_neg, pow_succ, mul_neg, Fin.prod_univ_succ, neg_mul,\n        ofReal_mul, neg_neg, B]\n      ring\n    \u00b7 exact h'f n h'n.le\n    \u00b7 exact hf.differentiable_iteratedFDeriv h'n\n    \u00b7 exact J\n\nend VectorFourier\n\nnamespace Real\nopen VectorFourier\n\nvariable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace \u211d V] [FiniteDimensional \u211d V]\n  [MeasurableSpace V] [BorelSpace V] {f : V \u2192 E}\n\n/-- The Fourier integral of the Fr\u00e9chet derivative of a function is obtained by multiplying the\nFourier integral of the original function by `2\u03c0I \u27eav, w\u27eb`. -/\ntheorem fourierIntegral_fderiv\n    (hf : Integrable f) (h'f : Differentiable \u211d f) (hf' : Integrable (fderiv \u211d f)) :\n    \ud835\udcd5 (fderiv \u211d f) = fourierSMulRight (-innerSL \u211d) (\ud835\udcd5 f) := by\n  rw [\u2190 innerSL_real_flip V]\n  exact VectorFourier.fourierIntegral_fderiv (innerSL \u211d) hf h'f hf'\n\n/-- The Fourier integral of the `n`-th derivative of a function is obtained by multiplying the\nFourier integral of the original function by `(2\u03c0I L w \u2b1d )^n`. -/\ntheorem fourierIntegral_iteratedFDeriv {N : \u2115\u221e} (hf : ContDiff \u211d N f)\n    (h'f : \u2200 (n : \u2115), n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f)) {n : \u2115} (hn : n \u2264 N) :\n    \ud835\udcd5 (iteratedFDeriv \u211d n f)\n      = (fun w \u21a6 fourierPowSMulRight (-innerSL \u211d) (\ud835\udcd5 f) w n) := by\n  rw [\u2190 innerSL_real_flip V]\n  exact VectorFourier.fourierIntegral_iteratedFDeriv (innerSL \u211d) hf h'f hn\n\n", "theoremStatement": "/-- The Fourier integral of the Fr\u00e9chet derivative of a function is obtained by multiplying the\nFourier integral of the original function by `2\u03c0I x`. -/\ntheorem fourierIntegral_deriv\n    {f : \u211d \u2192 E} (hf : Integrable f) (h'f : Differentiable \u211d f) (hf' : Integrable (deriv f)) :\n    \ud835\udcd5 (deriv f) = fun (x : \u211d) \u21a6 (2 * \u03c0 * I * x) \u2022 (\ud835\udcd5 f x) ", "theoremName": "Real.fourierIntegral_deriv", "fileCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "theoremCreated": {"commit": "c3dd1cff93aae1f610732113f84d025becc9053f", "date": "2024-04-08"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv.jsonl", "positionMetadata": {"lineInFile": 260, "tokenPositionInFile": 13433, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, 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"Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "Mathlib.Analysis.Fourier.FourierTransformDeriv"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  ext x\n  have I : Integrable (fun x \u21a6 fderiv \u211d f x) := by\n    simp_rw [\u2190 deriv_fderiv]\n    change Integrable (fun x \u21a6 ContinuousLinearMap.smulRightL _ _ _ 1 (deriv f x)) volume\n    apply ContinuousLinearMap.integrable_comp _ hf'\n  have : \ud835\udcd5 (deriv f) x = \ud835\udcd5 (fderiv \u211d f) x 1 := by\n    simp_rw [fourierIntegral_eq, deriv,\n      ContinuousLinearMap.integral_apply ((fourierIntegral_convergent_iff _).2 I)]\n    rfl\n  rw [this, fourierIntegral_fderiv hf h'f I]\n  have : x \u2022 \ud835\udcd5 f x = (x : \u2102) \u2022 \ud835\udcd5 f x := rfl\n  simp only [fourierSMulRight_apply, ContinuousLinearMap.neg_apply, innerSL_apply, smul_smul,\n    RCLike.inner_apply, conj_trivial, mul_one, neg_smul, smul_neg, neg_neg, neg_mul, this]", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 690}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\n", "theoremStatement": "lemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop ", "theoremName": "nnabla_bound_aux1", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "3e0eb1019735b3dfd0b1a5b2393498e8c3bcb040", "date": "2024-03-14"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 720, "tokenPositionInFile": 36675, "theoremPositionInFile": 70}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 46, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", 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"Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 170}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\n", "theoremStatement": "lemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) ", "theoremName": "summation_by_parts''", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": 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"Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", 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+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module sieve\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults\n\nnoncomputable section\n\nopen scoped BigOperators ArithmeticFunction\n\nopen Finset Real Nat Aux\n\nstructure Sieve where mk ::\n  support : Finset \u2115\n  prodPrimes : \u2115\n  prodPrimes_squarefree : Squarefree prodPrimes\n  weights : \u2115 \u2192 \u211d\n  weights_nonneg : \u2200 n : \u2115, 0 \u2264 weights n\n  totalMass : \u211d\n  nu : ArithmeticFunction \u211d\n  nu_mult : nu.IsMultiplicative\n  nu_pos_of_prime : \u2200 p : \u2115, p.Prime \u2192 p \u2223 prodPrimes \u2192 0 < nu p\n  nu_lt_one_of_prime : \u2200 p : \u2115, p.Prime \u2192 p \u2223 prodPrimes \u2192 nu p < 1\n\nattribute [arith_mult] Sieve.nu_mult\n\nnamespace Sieve\n\nvariable (s : Sieve)\nlocal notation3 \"\u03bd\" => Sieve.nu s\nlocal notation3 \"P\" => Sieve.prodPrimes s\nlocal notation3 \"a\" => Sieve.weights s\nlocal notation3 \"X\" => Sieve.totalMass s\nlocal notation3 \"A\" => Sieve.support s\n\n@[simp]\ndef multSum (d : \u2115) : \u211d :=\n  \u2211 n in A, if d \u2223 n then a n else 0\n\nlocal notation3 \"\ud835\udc9c\" => Sieve.multSum s\n\n-- A_d = \u03bd (d)/d X + R_d\n@[simp]\ndef rem (d : \u2115) : \u211d :=\n  \ud835\udc9c d - \u03bd d * X\n\nlocal notation3 \"R\" => Sieve.rem s\n\ndef siftedSum : \u211d :=\n  \u2211 d in A, if Coprime P d then a d else 0\n\n-- S = \u2211_{l|P, l\u2264\u221ay} g(l)\n-- Used in statement of the simple form of the selberg bound\ndef selbergTerms : ArithmeticFunction \u211d :=\n  s.nu.pmul (.prodPrimeFactors fun p =>  1 / (1 - \u03bd p))\n\nlocal notation3 \"g\" => Sieve.selbergTerms s\n\ndef selbergTerms_apply (d : \u2115) :\n    g d = \u03bd d * \u220f p in d.primeFactors, 1/(1 - \u03bd p) := by\n  unfold selbergTerms\n  by_cases h : d=0\n  \u00b7 rw [h]; simp\n  rw [ArithmeticFunction.pmul_apply, ArithmeticFunction.prodPrimeFactors_apply h]\n\ndef mainSum (\u03bcPlus : \u2115 \u2192 \u211d) : \u211d :=\n  \u2211 d in divisors P, \u03bcPlus d * \u03bd d\n\ndef errSum (\u03bcPlus : \u2115 \u2192 \u211d) : \u211d :=\n  \u2211 d in divisors P, |\u03bcPlus d| * |R d|\n\n\nsection UpperBoundSieve\n\ndef UpperMoebius (\u03bc_plus : \u2115 \u2192 \u211d) : Prop :=\n  \u2200 n : \u2115, (if n=1 then 1 else 0) \u2264 \u2211 d in n.divisors, \u03bc_plus d\n\nstructure UpperBoundSieve where mk ::\n  \u03bcPlus : \u2115 \u2192 \u211d\n  h\u03bcPlus : UpperMoebius \u03bcPlus\n\ninstance ubTo\u03bcPlus : CoeFun UpperBoundSieve fun _ => \u2115 \u2192 \u211d where coe ub := ub.\u03bcPlus\n\ndef LowerMoebius (\u03bcMinus : \u2115 \u2192 \u211d) : Prop :=\n  \u2200 n : \u2115, \u2211 d in n.divisors, \u03bcMinus d \u2264 (if n=1 then 1 else 0)\n\nstructure LowerBoundSieve where mk ::\n  \u03bcMinus : \u2115 \u2192 \u211d\n  h\u03bcMinus : LowerMoebius \u03bcMinus\n\ninstance lbTo\u03bcMinus : CoeFun LowerBoundSieve fun _ => \u2115 \u2192 \u211d where coe lb := lb.\u03bcMinus\n\nend UpperBoundSieve\n\nsection SieveLemmas\n\ntheorem prodPrimes_ne_zero : P \u2260 0 :=\n  Squarefree.ne_zero s.prodPrimes_squarefree\n\ntheorem squarefree_of_dvd_prodPrimes {d : \u2115} (hd : d \u2223 P) : Squarefree d :=\n  Squarefree.squarefree_of_dvd hd s.prodPrimes_squarefree\n\ntheorem squarefree_of_mem_divisors_prodPrimes {d : \u2115} (hd : d \u2208 divisors P) : Squarefree d := by\n  simp only [Nat.mem_divisors, Ne.def] at hd\n  exact Squarefree.squarefree_of_dvd hd.left s.prodPrimes_squarefree\n\ntheorem nu_pos_of_dvd_prodPrimes {d : \u2115} (hd : d \u2223 P) : 0 < \u03bd d := by\n  calc\n    0 < \u220f p in d.primeFactors, \u03bd p := by\n      apply prod_pos\n      intro p hpd\n      have hp_prime : p.Prime := by exact prime_of_mem_primeFactors hpd\n      have hp_dvd : p \u2223 P := (dvd_of_mem_primeFactors hpd).trans hd\n      exact s.nu_pos_of_prime p hp_prime hp_dvd\n    _ = \u03bd d := s.nu_mult.prod_factors_of_mult \u03bd (Squarefree.squarefree_of_dvd hd s.prodPrimes_squarefree)\n\ntheorem nu_ne_zero {d : \u2115} (hd : d \u2223 P) : \u03bd d \u2260 0 := by\n  apply _root_.ne_of_gt\n  exact nu_pos_of_dvd_prodPrimes s hd\n\ntheorem nu_ne_zero_of_mem_divisors_prodPrimes {d : \u2115} (hd : d \u2208 divisors P) : \u03bd d \u2260 0 := by\n  apply _root_.ne_of_gt\n  rw [mem_divisors] at hd\n  apply s.nu_pos_of_dvd_prodPrimes hd.left\n\ntheorem multSum_eq_main_err (d : \u2115) : s.multSum d = \u03bd d * X + R d := by\n  dsimp [rem]\n  ring\n\ndef delta (n : \u2115) : \u211d := if n=1 then 1 else 0\n\nlocal notation \"\u03b4\" => delta\n\ntheorem siftedSum_as_delta : s.siftedSum = \u2211 d in s.support, a d * \u03b4 (Nat.gcd P d) :=\n  by\n  dsimp only [siftedSum]\n  apply sum_congr rfl\n  intro d _\n  dsimp only [Nat.Coprime, delta] at *\n  rw [mul_ite_zero]\n  exact if_congr Iff.rfl (symm $ mul_one _) rfl\n\n-- 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"Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", 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"Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", 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"Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", 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"Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have hd_sq : Squarefree d := Squarefree.squarefree_of_dvd hdP s.prodPrimes_squarefree\n  calc\n    \u03bd d = \u220f p in d.primeFactors, \u03bd p :=\n      eq_comm.mp (s.nu_mult.prod_factors_of_mult \u03bd hd_sq)\n    _ < \u220f p in d.primeFactors, 1 := by\n      apply prod_lt_prod_of_nonempty\n      \u00b7 intro p hp\n        simp only [mem_primeFactors] at hp\n        apply s.nu_pos_of_prime p (by aesop) (hp.2.1.trans hdP)\n      \u00b7 intro p hpd; rw [mem_primeFactors_of_ne_zero hd_sq.ne_zero] at hpd\n        apply s.nu_lt_one_of_prime p hpd.left (hpd.2.trans hdP)\n      \u00b7 apply primeDivisors_nonempty _ <| (two_le_iff d).mpr \u27e8hd_sq.ne_zero, hd_ne_one\u27e9\n    _ = 1 := by\n      simp", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 654}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module selberg\n-/\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic\n\n/-!\n# The Selberg Sieve\n\nThis file proves `selberg_bound_simple`, the main theorem of the Selberg.\n-/\n\nnoncomputable section\n\nopen scoped BigOperators Classical Sieve\n\nopen Finset Real Nat Sieve.UpperBoundSieve ArithmeticFunction Sieve\n\nstructure SelbergSieve extends Sieve where mk ::\n  level : \u211d\n  one_le_level : 1 \u2264 level\n\nnamespace SelbergSieve\nset_option quotPrecheck false\n\nvariable (s : SelbergSieve)\nlocal notation3 \"\u03bd\" => Sieve.nu (toSieve s)\nlocal notation3 \"P\" => Sieve.prodPrimes (toSieve s)\nlocal notation3 \"a\" => Sieve.weights (toSieve s)\nlocal notation3 \"X\" => Sieve.totalMass (toSieve s)\nlocal notation3 \"R\" => Sieve.rem (toSieve s)  -- this one seems broken\nlocal notation3 \"g\" => Sieve.selbergTerms (toSieve s)\nlocal notation3 \"y\" => SelbergSieve.level s\nlocal notation3 \"hy\" => SelbergSieve.one_le_level s\n\n--set_option profiler true\n@[simp]\ndef selbergBoundingSum : \u211d :=\n  \u2211 l in divisors P, if l ^ 2 \u2264 y then g l else 0\n\nset_option quotPrecheck false\nlocal notation3 \"S\" => SelbergSieve.selbergBoundingSum s\n\ntheorem selbergBoundingSum_pos :\n    0 < S := by\n  dsimp only [selbergBoundingSum]\n  rw [\u2190 sum_filter]\n  apply sum_pos;\n  \u00b7 intro l hl\n    rw [mem_filter, mem_divisors] at hl\n    \u00b7 apply s.selbergTerms_pos _ (hl.1.1)\n  \u00b7 simp_rw [Finset.Nonempty, mem_filter]; use 1\n    constructor\n    \u00b7 apply one_mem_divisors.mpr s.prodPrimes_ne_zero\n    rw [cast_one, one_pow]\n    exact s.one_le_level\n\ntheorem selbergBoundingSum_ne_zero : S \u2260 0 := by\n  apply _root_.ne_of_gt\n  exact s.selbergBoundingSum_pos\n\ntheorem selbergBoundingSum_nonneg : 0 \u2264 S := _root_.le_of_lt s.selbergBoundingSum_pos\n\ndef selbergWeights : \u2115 \u2192 \u211d := fun d =>\n  if d \u2223 P then\n    (\u03bd d)\u207b\u00b9 * g d * \u03bc d * S\u207b\u00b9 *\n      \u2211 m in divisors P, if (d * m) ^ 2 \u2264 y \u2227 m.Coprime d then g m else 0\n  else 0\n\n-- This notation traditionally uses \u03bb, which is unavailable in lean\nset_option quotPrecheck false\nlocal notation3 \"\u03b3\" => SelbergSieve.selbergWeights s\n\ntheorem selbergWeights_eq_zero_of_not_dvd {d : \u2115} (hd : \u00ac d \u2223 P) :\n    \u03b3 d = 0 := by\n  rw [selbergWeights, if_neg hd]\n\ntheorem selbergWeights_eq_zero (d : \u2115) (hd : \u00acd ^ 2 \u2264 y) :\n    \u03b3 d = 0 := by\n  dsimp only [selbergWeights]\n  split_ifs with h\n  \u00b7 rw [mul_eq_zero_of_right _]\n    apply Finset.sum_eq_zero\n    refine fun m hm => if_neg ?_\n    intro hyp\n    have : (d^2:\u211d) \u2264 (d*m)^2 := by\n      norm_cast;\n      refine Nat.pow_le_pow_of_le_left ?h 2\n      exact Nat.le_mul_of_pos_right _ (Nat.pos_of_mem_divisors hm)\n    linarith [hyp.1]\n  \u00b7 rfl\n\n@[aesop safe]\ntheorem selbergWeights_mul_mu_nonneg (d : \u2115) (hdP : d \u2223 P) :\n    0 \u2264 \u03b3 d * \u03bc d :=\n  by\n  have := s.selbergBoundingSum_nonneg\n  dsimp only [selbergWeights]\n  rw [if_pos hdP]; rw [mul_assoc]\n  trans ((\u03bc d :\u211d)^2 * (\u03bd d)\u207b\u00b9 * g d * S\u207b\u00b9 * \u2211 m in divisors P,\n          if (d * m) ^ 2 \u2264 y \u2227 Coprime m d then g m else 0)\n  swap; apply le_of_eq; ring\n  apply mul_nonneg; apply div_nonneg; apply mul_nonneg; apply mul_nonneg\n  \u00b7 apply sq_nonneg\n  \u00b7 rw [inv_nonneg]\n    exact le_of_lt $ s.nu_pos_of_dvd_prodPrimes hdP\n  \u00b7 exact le_of_lt $ s.selbergTerms_pos d hdP\n  \u00b7 exact s.selbergBoundingSum_nonneg\n  apply sum_nonneg; intro m hm\n  split_ifs with h\n  \u00b7 exact le_of_lt $ s.selbergTerms_pos m (dvd_of_mem_divisors hm)\n  \u00b7 rfl\n\nlemma sum_mul_subst (k n: \u2115) {f : \u2115 \u2192 \u211d} (h : \u2200 l, l \u2223 n \u2192 \u00ac k \u2223 l \u2192 f l = 0) :\n      \u2211 l in n.divisors, f l\n    = \u2211 m in n.divisors, if k*m \u2223 n then f (k*m) else 0 := by\n  by_cases hn: n = 0\n  \u00b7 simp [hn]\n  by_cases hkn : k \u2223 n\n  swap\n  \u00b7 rw [sum_eq_zero, sum_eq_zero]\n    \u00b7 rintro m _\n      rw [if_neg]\n      rintro h\n      apply hkn\n      exact (Nat.dvd_mul_right k m).trans h\n    \u00b7 intro l hl; apply h l (dvd_of_mem_divisors hl)\n      apply fun hkl => hkn <| hkl.trans (dvd_of_mem_divisors hl)\n  trans (\u2211 l in n.divisors, \u2211 m in n.divisors, if l=k*m then f l else 0)\n  \u00b7 rw [sum_congr rfl]; intro l hl\n    by_cases hkl : k \u2223 l\n    swap\n    \u00b7 rw [h l (dvd_of_mem_divisors hl) hkl, sum_eq_zero];\n      intro m _; rw [ite_id]\n    rw [sum_eq_single (l/k)]\n    \u00b7 rw[if_pos]; rw [Nat.mul_div_cancel' hkl]\n    \u00b7 intro m _ hmlk\n      apply if_neg; revert hmlk; contrapose!; intro hlkm\n      rw [hlkm, mul_comm, Nat.mul_div_cancel];\n      apply Nat.pos_of_dvd_of_pos hkn (Nat.pos_of_ne_zero hn)\n    \u00b7 contrapose!; intro _\n      rw [mem_divisors]\n      exact \u27e8Trans.trans (Nat.div_dvd_of_dvd hkl) (dvd_of_mem_divisors hl), hn\u27e9\n  \u00b7 rw [sum_comm, sum_congr rfl]; intro m _\n    split_ifs with hdvd\n    \u00b7 rw [\u2190Aux.sum_intro]\n      simp only [mem_divisors, hdvd, ne_eq, hn, not_false_eq_true, and_self]\n    \u00b7 apply sum_eq_zero; intro l hl\n      apply if_neg;\n      rintro rfl\n      simp only [mem_divisors, ne_eq] at hl\n      exact hdvd hl.1\n\n--Important facts about the selberg weights\ntheorem selbergWeights_eq_dvds_sum (d : \u2115) :\n    \u03bd d * \u03b3 d =\n      S\u207b\u00b9 * \u03bc d *\n        \u2211 l in divisors P, if d \u2223 l \u2227 l ^ 2 \u2264 y then g l else 0 := by\n  by_cases h_dvd : d \u2223 P\n  swap\n  \u00b7 dsimp only [selbergWeights]; rw [if_neg h_dvd]\n    rw [sum_eq_zero]; ring\n    intro l hl; rw [mem_divisors] at hl\n    rw [if_neg]; push_neg; intro h\n    exfalso; exact h_dvd (dvd_trans h hl.left)\n  dsimp only [selbergWeights]\n  rw [if_pos h_dvd]\n  repeat rw [mul_sum]\n  -- change of variables l=m*d\n  apply symm\n  rw [sum_mul_subst d P]\n  apply sum_congr rfl\n  intro m hm\n  rw [mul_ite_zero, \u2190ite_and, mul_ite_zero, mul_ite_zero]\n  apply if_ctx_congr _ _ fun _ => rfl\n  \u00b7 rw [coprime_comm]\n    constructor\n    \u00b7 intro h\n      push_cast at h\n      exact \u27e8h.2.2, coprime_of_squarefree_mul $ Squarefree.squarefree_of_dvd h.1 s.prodPrimes_squarefree\u27e9\n    \u00b7 intro h\n      push_cast\n      exact \u27e8 Coprime.mul_dvd_of_dvd_of_dvd h.2 h_dvd (dvd_of_mem_divisors hm), Nat.dvd_mul_right d m, h.1\u27e9\n  \u00b7 intro h\n    trans ((\u03bd d)\u207b\u00b9 * (\u03bd d) * g d * \u03bc d / S * g m)\n    \u00b7 rw [inv_mul_cancel (s.nu_ne_zero h_dvd), s.selbergTerms_mult.map_mul_of_coprime\n        $ coprime_comm.mp h.2]\n      ring\n    ring\n  \u00b7 intro l _ hdl\n    rw [if_neg, mul_zero]\n    push_neg; intro h; contradiction\n\ntheorem selbergWeights_diagonalisation (l : \u2115) (hl : l \u2208 divisors P) :\n    (\u2211 d in divisors P, if l \u2223 d then \u03bd d * \u03b3 d else 0) =\n      if l ^ 2 \u2264 y then g l * \u03bc l * S\u207b\u00b9 else 0 := by\n  calc\n    (\u2211 d in divisors P, if l \u2223 d then \u03bd d * \u03b3 d else 0) =\n        \u2211 d in divisors P, \u2211 k in divisors P,\n          if l \u2223 d \u2227 d \u2223 k \u2227 k ^ 2 \u2264 y then g k * S\u207b\u00b9 * (\u03bc d:\u211d) else 0 := by\n      apply sum_congr rfl; intro d _\n      rw [selbergWeights_eq_dvds_sum, \u2190 boole_mul, mul_sum, mul_sum]\n      apply sum_congr rfl; intro k _\n      rw [mul_ite_zero, ite_zero_mul_ite_zero]\n      apply if_ctx_congr Iff.rfl _ (fun _ => rfl);\n      intro _; ring\n    _ = \u2211 k in divisors P, if k ^ 2 \u2264 y then\n            (\u2211 d in divisors P, if l \u2223 d \u2227 d \u2223 k then (\u03bc d:\u211d) else 0) * g k * S\u207b\u00b9\n          else 0 := by\n      rw [sum_comm]; apply sum_congr rfl; intro k _\n      apply symm\n      rw [\u2190 boole_mul, sum_mul, sum_mul, mul_sum, sum_congr rfl]\n      intro d _\n      rw [ite_zero_mul, ite_zero_mul, ite_zero_mul, one_mul, \u2190ite_and]\n      apply if_ctx_congr _ _ (fun _ => rfl)\n      \u00b7 tauto\n      intro _; ring\n    _ = if l ^ 2 \u2264 y then g l * \u03bc l * S\u207b\u00b9 else 0 := by\n      rw [Aux.sum_intro (f:=fun _ => if l^2 \u2264 y then g l * \u03bc l * S\u207b\u00b9 else 0) (divisors P) l hl]\n      apply sum_congr rfl; intro k hk\n      rw [Aux.moebius_inv_dvd_lower_bound_real s.prodPrimes_squarefree l _ (dvd_of_mem_divisors hk),\n        \u2190ite_and, ite_zero_mul, ite_zero_mul, \u2190 ite_and]\n      apply if_ctx_congr _ _ fun _ => rfl\n      rw [and_comm, eq_comm]; apply and_congr_right\n      intro heq; rw [heq]\n      intro h; rw[h.1]; ring\n\ndef selbergMuPlus : \u2115 \u2192 \u211d :=\n  Sieve.lambdaSquared \u03b3\n\nset_option quotPrecheck false\nlocal notation3 \"\u03bc\u207a\" => SelbergSieve.selbergMuPlus s\n\ntheorem weight_one_of_selberg : \u03b3 1 = 1 := by\n  dsimp only [selbergWeights]\n  rw [if_pos (one_dvd P), s.nu_mult.left, s.selbergTerms_mult.left]\n  -- rw [ArithmeticFunction.moebius_apply_one, Int.cast_one]\n  simp only [inv_one, mul_one, isUnit_one, IsUnit.squarefree, moebius_apply_of_squarefree,\n    cardFactors_one, _root_.pow_zero, Int.cast_one, selbergBoundingSum, cast_pow, one_mul,\n    coprime_one_right_eq_true, and_true, cast_one]\n  rw [inv_mul_cancel]\n  convert s.selbergBoundingSum_ne_zero\n\ntheorem selberg\u03bcPlus_eq_zero (d : \u2115) (hd : \u00acd \u2264 y) : \u03bc\u207a d = 0 :=\n  by\n  apply Sieve.lambdaSquared_eq_zero_of_support _ y _ d hd\n  apply s.selbergWeights_eq_zero\n\ndef selbergUbSieve : UpperBoundSieve :=\n  \u27e8\u03bc\u207a, Sieve.upperMoebius_of_lambda_sq \u03b3 (s.weight_one_of_selberg)\u27e9\n\n-- proved for general lambda squared sieves\ntheorem mainSum_eq_diag_quad_form :\n    s.mainSum \u03bc\u207a =\n      \u2211 l in divisors P,\n        1 / g l *\n          (\u2211 d in divisors P, if l \u2223 d then \u03bd d * \u03b3 d else 0) ^ 2 :=\n  by apply lambdaSquared_mainSum_eq_diag_quad_form\n\n\n/-- These two are in Mathlib per #10672 -/\ntheorem moebius_sq_eq_one_of_squarefree {l : \u2115} (hl : Squarefree l) : \u03bc l ^ 2 = 1 := by\n  rw [moebius_apply_of_squarefree hl, \u2190 pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]\n\ntheorem abs_moebius_eq_one_of_squarefree {l : \u2115} (hl : Squarefree l) : |\u03bc l| = 1 := by\n  simp only [moebius_apply_of_squarefree hl, abs_pow, abs_neg, abs_one, one_pow]\n\ntheorem selberg_bound_simple_mainSum :\n    s.mainSum \u03bc\u207a = S\u207b\u00b9 :=\n  by\n  rw [mainSum_eq_diag_quad_form]\n  trans (\u2211 l in divisors P, (if l ^ 2 \u2264 y then g l *  (S\u207b\u00b9) ^ 2 else 0))\n  \u00b7 apply sum_congr rfl; intro l hl\n    rw [s.selbergWeights_diagonalisation l hl, ite_pow, zero_pow, mul_ite_zero]\n    apply if_congr Iff.rfl _ rfl\n    trans (1/g l * g l * g l * (\u03bc l:\u211d)^2  * (S\u207b\u00b9) ^ 2)\n    \u00b7 ring\n    norm_cast; rw [moebius_sq_eq_one_of_squarefree $ s.squarefree_of_mem_divisors_prodPrimes hl]\n    rw [one_div_mul_cancel $ _root_.ne_of_gt $ s.selbergTerms_pos l $ dvd_of_mem_divisors hl]\n    ring\n    linarith\n  conv => {lhs; congr; {skip}; {ext i; rw [\u2190 ite_zero_mul]}}\n  dsimp only [selbergBoundingSum]\n  rw [\u2190sum_mul, sq, \u2190mul_assoc, mul_inv_cancel]; ring\n  apply _root_.ne_of_gt; apply selbergBoundingSum_pos;\n\nlemma eq_gcd_mul_of_dvd_of_coprime {k d m :\u2115} (hkd : k \u2223 d) (hmd : Coprime m d) (hk : k \u2260 0) :\n    k = d.gcd (k*m) := by\n  cases' hkd with r hr\n  have hrdvd : r \u2223 d := by use k; rw [mul_comm]; exact hr\n  apply symm; rw [hr, Nat.gcd_mul_left, mul_eq_left\u2080 hk, Nat.gcd_comm]\n  apply Coprime.coprime_dvd_right hrdvd hmd\n\nprivate lemma _helper {k m d :\u2115} (hkd : k \u2223 d) (hk: k \u2208 divisors P) (hm: m \u2208 divisors P):\n    k * m \u2223 P \u2227 k = Nat.gcd d (k * m) \u2227 (k * m) ^ 2 \u2264 y \u2194\n    (k * m) ^ 2 \u2264 y \u2227 Coprime m d := by\n  constructor\n  \u00b7 intro h\n    constructor\n    \u00b7 exact h.2.2\n    \u00b7 cases' hkd with r hr\n      rw [hr, Nat.gcd_mul_left, eq_comm, mul_eq_left\u2080 (by rintro rfl; simp at hk \u22a2)] at h\n      rw [hr, coprime_comm]; apply Coprime.mul\n      apply coprime_of_squarefree_mul $ Squarefree.squarefree_of_dvd h.1 s.prodPrimes_squarefree\n      exact h.2.1\n  \u00b7 intro h\n    constructor\n    \u00b7 apply Nat.Coprime.mul_dvd_of_dvd_of_dvd\n      rw [coprime_comm]; exact Coprime.coprime_dvd_right hkd h.2\n      exact dvd_of_mem_divisors hk; exact dvd_of_mem_divisors hm\n    constructor\n    \u00b7 exact eq_gcd_mul_of_dvd_of_coprime hkd h.2 (by rintro rfl; simp at hk \u22a2)\n    \u00b7 exact h.1\n\n", "theoremStatement": "theorem selbergBoundingSum_ge {d : \u2115} (hdP : d \u2223 P) :\n    S \u2265 \u03b3 d * \u2191(\u03bc d) * S ", "theoremName": "SelbergSieve.selbergBoundingSum_ge", "fileCreated": {"commit": 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"Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  calc\n  _ = (\u2211 k in divisors P, \u2211 l in divisors P, if k = d.gcd l \u2227 l ^ 2 \u2264 y then g l else 0) := by\n    dsimp only [selbergBoundingSum]\n    rw [sum_comm, sum_congr rfl]; intro l _\n    simp_rw [ite_and]\n    rw [\u2190Aux.sum_intro]\n    \u00b7 rw [mem_divisors]\n      exact \u27e8(Nat.gcd_dvd_left d l).trans (hdP), s.prodPrimes_ne_zero\u27e9\n  _ = (\u2211 k in divisors P,\n          if k \u2223 d then\n            g k * \u2211 m in divisors P, if (k * m) ^ 2 \u2264 y \u2227 m.Coprime d then g m else 0\n          else 0) := by\n    apply sum_congr rfl; intro k hk\n    rw [mul_sum]\n    split_ifs with hkd\n    swap\n    \u00b7 rw [sum_eq_zero]; intro l _\n      rw [if_neg]\n      push_neg; intro h; exfalso\n      rw [h] at hkd\n      exact hkd $ Nat.gcd_dvd_left d l\n    rw [sum_mul_subst k P, sum_congr rfl]; intro m hm\n    rw [mul_ite_zero, \u2190 ite_and]\n    apply if_ctx_congr _ _ fun _ => rfl\n    \u00b7 exact_mod_cast s._helper hkd hk hm\n    \u00b7 intro h;\n      apply s.selbergTerms_mult.2\n      rw [coprime_comm]; apply h.2.coprime_dvd_right hkd\n    \u00b7 intro l _ hkl; apply if_neg\n      push_neg; intro h; exfalso\n      rw [h] at hkl; exact hkl (Nat.gcd_dvd_right d l)\n  _ \u2265 (\u2211 k in divisors P, if k \u2223 d\n          then g k * \u2211 m in divisors P, if (d * m) ^ 2 \u2264 y \u2227 m.Coprime d then g m else 0\n          else 0 ) := by\n    apply sum_le_sum; intro k _\n    split_ifs with hkd\n    swap; rfl\n    apply mul_le_mul le_rfl _ _ (le_of_lt $ s.selbergTerms_pos k $ Trans.trans hkd hdP)\n    apply sum_le_sum; intro m hm\n    split_ifs with h h' h'\n    \u00b7 rfl\n    \u00b7 exfalso; apply h'\n      refine \u27e8?_, h.2\u27e9\n      \u00b7 trans ((d*m)^2:\u211d)\n        \u00b7 norm_cast; gcongr\n          refine Nat.le_of_dvd ?_ hkd\n          apply Nat.pos_of_ne_zero; apply ne_zero_of_dvd_ne_zero s.prodPrimes_ne_zero hdP\n        exact h.1\n    \u00b7 refine le_of_lt $ s.selbergTerms_pos m $ dvd_of_mem_divisors hm\n    \u00b7 rfl\n    apply sum_nonneg; intro m hm\n    split_ifs\n    \u00b7 apply le_of_lt $ s.selbergTerms_pos m $ dvd_of_mem_divisors hm\n    \u00b7 rfl\n  _ = _ := by\n    conv => enter [1, 2, k]; rw [\u2190 ite_zero_mul]\n    rw [\u2190sum_mul, s.conv_selbergTerms_eq_selbergTerms_mul_nu hdP]\n    trans (S * S\u207b\u00b9 * (\u03bc d:\u211d)^2 * (\u03bd d)\u207b\u00b9 * g d * (\u2211 m in divisors P, if (d*m) ^ 2 \u2264 y \u2227 Coprime m d then g m else 0))\n    \u00b7 rw [mul_inv_cancel, \u2190Int.cast_pow, moebius_sq_eq_one_of_squarefree]\n      ring\n      exact Squarefree.squarefree_of_dvd hdP s.prodPrimes_squarefree\n      exact _root_.ne_of_gt $ s.selbergBoundingSum_pos\n    dsimp only [selbergWeights]; rw [if_pos hdP]\n    ring", "proofType": "tactic", "proofLengthLines": 64, "proofLengthTokens": 2456}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n", "theoremStatement": "@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u ", "theoremName": "nabla_mul", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "d0e2fa4e050c2209cfa69b9be857ee4dd7caefa8", "date": "2024-03-11"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": 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"Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  ext n ; simp [nabla, mul_sub]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 37}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\nlemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]\n\nlemma sum_eq_int_deriv_aux_lt {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_lt_kpOne : b < k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k := Int.floor_eq_iff.mpr \u27e8by linarith [ha.1, ha.2], by linarith\u27e9\n  simp only [flb_eq_k, gt_iff_lt, lt_add_iff_pos_right, zero_lt_one, Finset.Icc_eq_empty_of_lt,\n    Finset.sum_empty]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont]\n  have : Finset.Ioc k k = {} := by simp only [ge_iff_le, le_refl, Finset.Ioc_eq_empty_of_le]\n  simp only [this, Finset.sum_empty, one_div]; ring_nf\n\nlemma sum_eq_int_deriv_aux1 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  by_cases h : b = k + 1\n  \u00b7 exact sum_eq_int_deriv_aux_eq h \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact sum_eq_int_deriv_aux_lt ha (Ne.lt_of_le h b_le_kpOne) \u03c6Diff deriv\u03c6Cont\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv_aux]\\label{sum_eq_int_deriv_aux}\\lean{sum_eq_int_deriv_aux}\\leanok\n  Let $k \\le a < b\\le k+1$, with $k$ an integer, and let $\\phi$ be continuously differentiable on\n  $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma sum_eq_int_deriv_aux {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  have fl_a_eq_k : \u230aa\u230b = k := Int.floor_eq_iff.mpr \u27e8ha.1, by linarith [ha.2]\u27e9\n  convert sum_eq_int_deriv_aux1 ha b_le_kpOne \u03c6Diff deriv\u03c6Cont using 2\n  \u00b7 rw [fl_a_eq_k]\n  \u00b7 congr\n  \u00b7 apply intervalIntegral.integral_congr_ae\n    have : \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.volume, x \u2260 b := by\n      convert Countable.ae_not_mem (s := {b}) (by simp) (\u03bc := MeasureTheory.volume) using 1\n    filter_upwards [this]\n    intro x x_ne_b hx\n    rw [uIoc_of_le ha.2.le, mem_Ioc] at hx\n    congr\n    exact Int.floor_eq_iff.mpr \u27e8by linarith [ha.1], by have := Ne.lt_of_le x_ne_b hx.2; linarith\u27e9\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\n-- Thanks to Arend Mellendijk\n\nlemma interval_induction_aux_int (n : \u2115) : \u2200 (P : \u211d \u2192 \u211d \u2192 Prop)\n    (_ : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (_ : \u2200 (a : \u211d) (k : \u2124) (c : \u211d), a < k \u2192 k < c \u2192 P a k \u2192 P k c \u2192 P a c)\n    (a b : \u211d) (_ : a < b) (_ : n = \u230ab\u230b - \u230aa\u230b),\n    P a b := by\n  induction n using Nat.case_strong_induction_on with\n  | hz =>\n    intro P base _ a b hab hn\n    apply base a b \u230aa\u230b (Int.floor_le a) hab\n    rw [(by simp only [CharP.cast_eq_zero] at hn; linarith : \u230aa\u230b = \u230ab\u230b)]\n    exact (Int.lt_floor_add_one b).le\n  | hi n ih =>\n    intro P base step a b _ hn\n    have Pa : P a (\u230aa\u230b + 1) :=\n      base a (\u230aa\u230b + 1) \u230aa\u230b (Int.floor_le a) (Int.lt_floor_add_one a) (le_of_eq rfl)\n    by_cases b_le_flaP1 : b = \u230aa\u230b + 1\n    \u00b7 rwa [b_le_flaP1]\n    have flaP1_lt_b : \u230aa\u230b + 1 < b := by\n      simp only [Nat.cast_succ] at hn\n      have : (\u230aa\u230b : \u211d) + 1 \u2264 \u230ab\u230b := by exact_mod_cast (by linarith)\n      exact Ne.lt_of_le (id (Ne.symm b_le_flaP1)) (by linarith [Int.floor_le b] : \u230aa\u230b + 1 \u2264 b)\n    have Pfla_b : P (\u230aa\u230b + 1) b := by\n      apply ih n (le_of_eq rfl) P base step (\u230aa\u230b + 1) b flaP1_lt_b\n      simp only [Int.floor_add_one, Int.floor_intCast, Nat.cast_succ] at hn \u22a2\n      linarith\n    refine step a (\u230aa\u230b + 1) b ?_ (by exact_mod_cast flaP1_lt_b) (by exact_mod_cast Pa)\n      (by exact_mod_cast Pfla_b)\n    have := Int.lt_floor_add_one a\n    exact_mod_cast this\n\nlemma interval_induction (P : \u211d \u2192 \u211d \u2192 Prop)\n    (base : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (step : \u2200 (a : \u211d) (k : \u2124) (b : \u211d), a < k \u2192 k < b \u2192 P a k \u2192 P k b \u2192 P a b)\n    (a b : \u211d) (hab : a < b) : P a b := by\n  set n := \u230ab\u230b - \u230aa\u230b with hn\n  clear_value n\n  have : 0 \u2264 n := by simp only [hn, sub_nonneg, ge_iff_le, Int.floor_le_floor _ _ (hab.le)]\n  lift n to \u2115 using this\n  exact interval_induction_aux_int n P base step a b hab hn\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv]\\label{sum_eq_int_deriv}\\lean{sum_eq_int_deriv}\\leanok\n  Let $a < b$, and let $\\phi$ be continuously differentiable on $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\n/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {\u03b1 : Type*} [LinearOrder \u03b1] [LocallyFiniteOrder \u03b1]\n    {a b c: \u03b1} [DecidableEq \u03b1] (hb : b \u2208 Icc a c) : Ioc a b = Ioc a c \\ Ioc b c := by\n  ext x\n  simp only [mem_Ioc, mem_sdiff, not_and, not_le]\n  constructor\n  \u00b7 refine fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8\u27e8h\u2081, le_trans h\u2082 (mem_Icc.mp hb).2\u27e9, by contrapose! h\u2082; exact h\u2082.1\u27e9\n  \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8h\u2081.1, by contrapose! h\u2082; exact \u27e8h\u2082, h\u2081.2\u27e9\u27e9\n\n-- In Ya\u00ebl Dillies's API (https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Esum_add_adjacent_intervals/near/430127101)\nlemma Finset.sum_Ioc_add_sum_Ioc {a b c : \u2124} (f : \u2124 \u2192 \u2102) (hb : b \u2208 Icc a c):\n    (\u2211 n in Finset.Ioc a b, f n) + (\u2211 n in Finset.Ioc b c, f n) = \u2211 n in Finset.Ioc a c, f n := by\n  convert Finset.sum_sdiff (s\u2081 := Finset.Ioc b c) (s\u2082 := Finset.Ioc a c) ?_\n  \u00b7 exact Finset.Ioc_diff_Ioc hb\n  \u00b7 exact Finset.Ioc_subset_Ioc (mem_Icc.mp hb).1 (by rfl)\n\nlemma integrability_aux\u2080 {a b : \u211d} :\n    \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.Measure.restrict MeasureTheory.volume [[a, b]],\n      \u2016(\u230ax\u230b : \u2102)\u2016 \u2264 max \u2016a\u2016 \u2016b\u2016 + 1 := by\n  apply (MeasureTheory.ae_restrict_iff' measurableSet_Icc).mpr\n  refine MeasureTheory.ae_of_all _ (fun x hx \u21a6 ?_)\n  simp only [inf_le_iff, le_sup_iff, mem_Icc] at hx\n  simp only [norm_int, Real.norm_eq_abs]\n  have : |x| \u2264 max |a| |b| := by\n    cases' hx.1 with x_ge_a x_ge_b <;> cases' hx.2 with x_le_a x_le_b\n    \u00b7 rw [(by linarith : x = a)]; apply le_max_left\n    \u00b7 apply abs_le_max_abs_abs x_ge_a x_le_b\n    \u00b7 rw [max_comm]; apply abs_le_max_abs_abs x_ge_b x_le_a\n    \u00b7 rw [(by linarith : x = b)]; apply le_max_right\n  cases' abs_cases x with hx hx\n  \u00b7 rw [_root_.abs_of_nonneg <| by exact_mod_cast Int.floor_nonneg.mpr hx.2]\n    apply le_trans (Int.floor_le x) <| le_trans (hx.1 \u25b8 this) (by simp)\n  \u00b7 rw [_root_.abs_of_nonpos <| by exact_mod_cast Int.floor_nonpos hx.2.le]\n    linarith [(Int.lt_floor_add_one x).le]\n\nlemma integrability_aux\u2081 {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (\u230ax\u230b : \u2102)) MeasureTheory.volume a b := by\n  rw [intervalIntegrable_iff']\n  apply MeasureTheory.Measure.integrableOn_of_bounded ?_ ?_ integrability_aux\u2080\n  \u00b7 simp only [Real.volume_interval, ne_eq, ENNReal.ofReal_ne_top, not_false_eq_true]\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply Measurable.comp (by exact fun \u2983t\u2984 _ \u21a6 trivial) Int.measurable_floor\n\nlemma integrability_aux\u2082 {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (1 : \u2102) / 2 - x) MeasureTheory.volume a b :=\n  ContinuousOn.intervalIntegrable <| Continuous.continuousOn (by continuity)\n\nlemma integrability_aux {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (\u230ax\u230b : \u2102) + 1 / 2 - x) MeasureTheory.volume a b := by\n  convert integrability_aux\u2081.add integrability_aux\u2082 using 2; ring\n\nlemma uIcc_subsets {a b c : \u211d} (hc : c \u2208 Icc a b) :\n    [[a, c]] \u2286 [[a, b]] \u2227 [[c, b]] \u2286 [[a, b]] := by\n  constructor <;> rw [uIcc_of_le ?_, uIcc_of_le ?_]\n  any_goals apply Icc_subset_Icc\n  all_goals linarith [hc.1, hc.2]\n\nlemma sum_eq_int_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (a_lt_b : a < b)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n      (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n        - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  let P := fun a\u2081 b\u2081 \u21a6 (\u2200 x \u2208 [[a\u2081, b\u2081]], HasDerivAt \u03c6 (deriv \u03c6 x) x) \u2192\n    (ContinuousOn (deriv \u03c6) [[a\u2081, b\u2081]]) \u2192\n    \u2211 n in Finset.Ioc \u230aa\u2081\u230b \u230ab\u2081\u230b, \u03c6 n =\n    (\u222b x in a\u2081..b\u2081, \u03c6 x) + (\u230ab\u2081\u230b + 1 / 2 - b\u2081) * \u03c6 b\u2081 - (\u230aa\u2081\u230b + 1 / 2 - a\u2081) * \u03c6 a\u2081\n      - \u222b x in a\u2081..b\u2081, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x\n  apply interval_induction P ?base ?step a b a_lt_b \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact fun _ _ _ k\u2081_le_a\u2081 a\u2081_lt_b\u2081 b\u2081_le_k\u2081 \u03c6Diff\u2081 deriv\u03c6Cont\u2081 \u21a6\n      sum_eq_int_deriv_aux \u27e8k\u2081_le_a\u2081, a\u2081_lt_b\u2081\u27e9 b\u2081_le_k\u2081 \u03c6Diff\u2081 deriv\u03c6Cont\u2081\n  \u00b7 intro a\u2081 k\u2081 b\u2081 a\u2081_lt_k\u2081 k\u2081_lt_b\u2081 ih\u2081 ih\u2082 \u03c6Diff\u2081 deriv\u03c6Cont\u2081\n    have subs := uIcc_subsets \u27e8a\u2081_lt_k\u2081.le, k\u2081_lt_b\u2081.le\u27e9\n    have s\u2081 := ih\u2081 (fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.1 hx) <| deriv\u03c6Cont\u2081.mono subs.1\n    have s\u2082 := ih\u2082 (fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.2 hx) <| deriv\u03c6Cont\u2081.mono subs.2\n    convert Mathlib.Tactic.LinearCombination.add_pf s\u2081 s\u2082 using 1\n    \u00b7 rw [\u2190 Finset.sum_Ioc_add_sum_Ioc]\n      simp only [Finset.mem_Icc, Int.floor_intCast, Int.le_floor]\n      exact \u27e8Int.cast_le.mp <| le_trans (Int.floor_le a\u2081) a\u2081_lt_k\u2081.le, k\u2081_lt_b\u2081.le\u27e9\n    \u00b7 set I\u2081 := \u222b (x : \u211d) in a\u2081..b\u2081, \u03c6 x\n      set I\u2082 := \u222b (x : \u211d) in a\u2081..k\u2081, \u03c6 x\n      set I\u2083 := \u222b (x : \u211d) in k\u2081..b\u2081, \u03c6 x\n      set J\u2081 := \u222b (x : \u211d) in a\u2081..b\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      set J\u2082 := \u222b (x : \u211d) in a\u2081..k\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      set J\u2083 := \u222b (x : \u211d) in k\u2081..b\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      have hI : I\u2082 + I\u2083 = I\u2081 := by\n        apply intervalIntegral.integral_add_adjacent_intervals <;>\n        apply ContinuousOn.intervalIntegrable\n        \u00b7 exact HasDerivAt.continuousOn <| fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.1 hx\n        \u00b7 exact HasDerivAt.continuousOn <| fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.2 hx\n      have hJ : J\u2082 + J\u2083 = J\u2081 := by\n        apply intervalIntegral.integral_add_adjacent_intervals <;>\n        apply IntervalIntegrable.mul_continuousOn\n        any_goals apply integrability_aux\n        \u00b7 exact deriv\u03c6Cont\u2081.mono subs.1\n        \u00b7 exact deriv\u03c6Cont\u2081.mono subs.2\n      rw [\u2190 hI, \u2190 hJ]; ring\n/-%%\n\\begin{proof}\\uses{sum_eq_int_deriv_aux}\\leanok\n  Apply Lemma \\ref{sum_eq_int_deriv_aux} in blocks of length $\\le 1$.\n\\end{proof}\n%%-/\n\nlemma xpos_of_uIcc {a b : \u2115} (ha : a \u2208 Ioo 0 b) {x : \u211d} (x_in : x \u2208 [[(a : \u211d), b]]) :\n    0 < x := by\n  rw [uIcc_of_le (by exact_mod_cast ha.2.le), mem_Icc] at x_in\n  linarith [(by exact_mod_cast ha.1 : (0 : \u211d) < a)]\n\nlemma neg_s_ne_neg_one {s : \u2102} (s_ne_one : s \u2260 1) : -s \u2260 -1 := fun hs \u21a6 s_ne_one <| neg_inj.mp hs\n\nlemma ZetaSum_aux1\u2081 {a b : \u2115} {s : \u2102} (s_ne_one : s \u2260 1) (ha : a \u2208 Ioo 0 b) :\n    (\u222b (x : \u211d) in a..b, 1 / (x : \u2102) ^ s) =\n    (b ^ (1 - s) - a ^ (1 - s)) / (1 - s) := by\n  convert integral_cpow (a := a) (b := b) (r := -s) ?_ using 1\n  \u00b7 refine intervalIntegral.integral_congr fun x hx \u21a6 one_div_cpow_eq ?_\n    exact (xpos_of_uIcc ha hx).ne'\n  \u00b7 norm_cast; rw [(by ring : -s + 1 = 1 - s)]\n  \u00b7 right; refine \u27e8neg_s_ne_neg_one s_ne_one, ?_\u27e9\n    exact fun hx \u21a6 (lt_self_iff_false 0).mp <| xpos_of_uIcc ha hx\n\nlemma ZetaSum_aux1\u03c6Diff {s : \u2102} {x : \u211d} (xpos : 0 < x) :\n    HasDerivAt (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) (deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) x) x := by\n  apply hasDerivAt_deriv_iff.mpr <| DifferentiableAt.div (differentiableAt_const _) ?_ ?_\n  \u00b7 exact Real.differentiableAt_cpow_const_of_ne s xpos\n  \u00b7 simp [cpow_eq_zero_iff, xpos.ne']\n\nlemma ZetaSum_aux1\u03c6deriv {s : \u2102} (s_ne_zero : s \u2260 0) {x : \u211d} (xpos : 0 < x) :\n    deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) x = (fun (x : \u211d) \u21a6 -s * (x : \u2102) ^ (-(s + 1))) x := by\n  let r := -s - 1\n  have r_add1_ne_zero : r + 1 \u2260 0 := fun hr \u21a6 by simp [neg_ne_zero.mpr s_ne_zero, r] at hr\n  have r_ne_neg1 : r \u2260 -1 := fun hr \u21a6 (hr \u25b8 r_add1_ne_zero) <| by norm_num\n  have hasDeriv := hasDerivAt_ofReal_cpow xpos.ne' r_ne_neg1\n  have := hasDeriv.deriv \u25b8 deriv_const_mul (-s) (hasDeriv).differentiableAt\n  convert this using 2\n  \u00b7 ext y\n    by_cases y_zero : (y : \u2102) = 0\n    \u00b7 simp only [y_zero, ofReal_zero, ne_eq, s_ne_zero, not_false_eq_true, zero_cpow, div_zero,\n      r_add1_ne_zero, zero_div, mul_zero]\n    \u00b7 have : (y : \u2102) ^ s \u2260 0 := fun hy \u21a6 y_zero ((cpow_eq_zero_iff _ _).mp hy).1\n      field_simp [r, mul_assoc, \u2190 Complex.cpow_add]\n  \u00b7 ring_nf\n\nlemma ZetaSum_aux1deriv\u03c6Cont {s : \u2102} (s_ne_zero : s \u2260 0) {a b : \u2115} (ha : a \u2208 Ioo 0 b) :\n    ContinuousOn (deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s)) [[a, b]] := by\n  have : EqOn _ (fun (t : \u211d) \u21a6 -s * (t : \u2102) ^ (-(s + 1))) [[a, b]] :=\n    fun x hx \u21a6 ZetaSum_aux1\u03c6deriv s_ne_zero <| xpos_of_uIcc ha hx\n  refine ContinuousOn.congr ?_ this\n  refine (ContinuousOn.cpow_const continuous_ofReal.continuousOn ?_).const_smul (c := -s)\n  exact fun x hx \u21a6 ofReal_mem_slitPlane.mpr <| xpos_of_uIcc ha hx\n\n/-%%\n\\begin{lemma}[ZetaSum_aux1]\\label{ZetaSum_aux1}\\lean{ZetaSum_aux1}\\leanok\n  Let $0 < a < b$ be natural numbers and $s\\in \\C$ with $s \\ne 1$ and $s \\ne 0$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\frac{1}{n^s} =  \\frac{b^{1-s} - a^{1-s}}{1-s} + \\frac{b^{-s}-a^{-s}}{2} + s \\int_a^b \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma ZetaSum_aux1 {a b : \u2115} {s : \u2102} (s_ne_one : s \u2260 1) (s_ne_zero : s \u2260 0) (ha : a \u2208 Ioo 0 b) :\n    \u2211 n in Finset.Ioc (a : \u2124) b, 1 / (n : \u2102) ^ s =\n    (b ^ (1 - s) - a ^ (1 - s)) / (1 - s) + 1 / 2 * (1 / b ^ (s)) - 1 / 2 * (1 / a ^ s)\n      + s * \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1)) := by\n  let \u03c6 := fun (x : \u211d) \u21a6 1 / (x : \u2102) ^ s\n  let \u03c6' := fun (x : \u211d) \u21a6 -s * (x : \u2102) ^ (-(s + 1))\n  have xpos : \u2200 x \u2208 [[(a : \u211d), b]], 0 < x := fun x hx \u21a6 xpos_of_uIcc ha hx\n  have \u03c6Diff : \u2200 x \u2208 [[(a : \u211d), b]], HasDerivAt \u03c6 (deriv \u03c6 x) x := fun x hx \u21a6 ZetaSum_aux1\u03c6Diff (xpos x hx)\n  have \u03c6deriv : \u2200 x \u2208 [[(a : \u211d), b]], deriv \u03c6 x = \u03c6' x := by\n    exact fun x hx \u21a6 ZetaSum_aux1\u03c6deriv s_ne_zero (xpos x hx)\n  have deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]] := ZetaSum_aux1deriv\u03c6Cont s_ne_zero ha\n  convert sum_eq_int_deriv (by exact_mod_cast ha.2) \u03c6Diff deriv\u03c6Cont using 1\n  \u00b7 congr <;> simp only [Int.floor_natCast]\n  \u00b7 rw [Int.floor_natCast, Int.floor_natCast, \u2190 intervalIntegral.integral_const_mul]\n    simp_rw [mul_div, \u2190 mul_div, ZetaSum_aux1\u2081 s_ne_one ha]\n    conv => rhs; rw [sub_eq_add_neg]\n    congr; any_goals norm_cast; simp only [one_div, add_sub_cancel_left]\n    rw [\u2190 intervalIntegral.integral_neg, intervalIntegral.integral_congr]\n    intro x hx; simp_rw [\u03c6deriv x hx, \u03c6']; ring_nf\n/-%%\n\\begin{proof}\\uses{sum_eq_int_deriv}\\leanok\n  Apply Lemma \\ref{sum_eq_int_deriv} to the function $x \\mapsto x^{-s}$.\n\\end{proof}\n%%-/\n\nlemma ZetaSum_aux1_1' {a b x : \u211d} (apos : 0 < a) (hx : x \u2208 Icc a b)\n    : 0 < x := lt_of_lt_of_le apos hx.1\n\nlemma ZetaSum_aux1_1 {a b x : \u211d} (apos : 0 < a) (a_lt_b : a < b) (hx : x \u2208 [[a,b]])\n    : 0 < x :=  lt_of_lt_of_le apos (uIcc_of_le a_lt_b.le \u25b8 hx).1\n\nlemma ZetaSum_aux1_2 {a b : \u211d} {c : \u211d} (apos : 0 < a) (a_lt_b : a < b)\n    (h : c \u2260 0 \u2227 0 \u2209 [[a, b]]) :\n    \u222b (x : \u211d) in a..b, 1 / x ^ (c+1) = (a ^ (-c) - b ^ (-c)) / c := by\n  rw [(by ring : (a ^ (-c) - b ^ (-c)) / c = (b ^ (-c) - a ^ (-c)) / (-c))]\n  have := integral_rpow (a := a) (b := b) (r := -c-1) (Or.inr \u27e8by simp [h.1], h.2\u27e9)\n  simp only [sub_add_cancel] at this\n  rw [\u2190 this]\n  apply intervalIntegral.integral_congr\n  intro x hx\n  have : 0 \u2264 x := (ZetaSum_aux1_1 apos a_lt_b hx).le\n  simp [div_rpow_eq_rpow_neg _ _ _ this, sub_eq_add_neg, add_comm]\n\nlemma ZetaSum_aux1_3a (x : \u211d) : -(1/2) < \u230a x \u230b + 1/2 - x := by\n  norm_num [\u2190 add_assoc]; linarith [sub_pos_of_lt (Int.lt_floor_add_one x)]\n\nlemma ZetaSum_aux1_3b (x : \u211d) : \u230ax\u230b + 1/2 - x \u2264 1/2 := by\n  ring_nf; exact add_le_of_nonpos_right <| sub_nonpos.mpr (Int.floor_le x)\n\nlemma ZetaSum_aux1_3 (x : \u211d) : |(\u230ax\u230b + 1/2 - x)| \u2264 1/2 :=\n  abs_le.mpr \u27e8le_of_lt (ZetaSum_aux1_3a x), ZetaSum_aux1_3b x\u27e9\n\nlemma ZetaSum_aux1_4' (x : \u211d) (hx : 0 < x) (s : \u2102) :\n      \u2016(\u230ax\u230b + 1 / 2 - (x : \u211d)) / (x : \u2102) ^ (s + 1)\u2016 =\n      |\u230ax\u230b + 1 / 2 - x| / x ^ ((s + 1).re) := by\n  simp [map_div\u2080, abs_ofReal, Complex.abs_cpow_eq_rpow_re_of_pos hx, \u2190 abs_ofReal]\n\nlemma ZetaSum_aux1_4 {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} :\n  \u222b (x : \u211d) in a..b, \u2016(\u2191\u230ax\u230b + (1 : \u211d) / 2 - \u2191x) / (x : \u2102) ^ (s + 1)\u2016 =\n    \u222b (x : \u211d) in a..b, |\u230ax\u230b + 1 / 2 - x| / x ^ (s + 1).re := by\n  apply intervalIntegral.integral_congr\n  exact fun x hx \u21a6 ZetaSum_aux1_4' x (ZetaSum_aux1_1 apos a_lt_b hx) s\n\nlemma ZetaSum_aux1_5a {a b : \u211d} (apos : 0 < a) {s : \u2102} (x : \u211d)\n  (h : x \u2208 Icc a b) : |\u2191\u230ax\u230b + 1 / 2 - x| / x ^ (s.re + 1) \u2264 1 / x ^ (s.re + 1) := by\n  apply div_le_div_of_nonneg_right _ _\n  \u00b7 exact le_trans (ZetaSum_aux1_3 x) (by norm_num)\n  \u00b7 apply Real.rpow_nonneg <| le_of_lt (ZetaSum_aux1_1' apos h)\n\nlemma ZetaSum_aux1_5b {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} (\u03c3pos : 0 < s.re) :\n  IntervalIntegrable (fun u \u21a6 1 / u ^ (s.re + 1)) MeasureTheory.volume a b := by\n  apply ContinuousOn.intervalIntegrable_of_Icc (le_of_lt a_lt_b) _\n  apply ContinuousOn.div continuousOn_const\n  \u00b7 refine ContinuousOn.rpow_const continuousOn_id ?_\n    exact fun x hx \u21a6 Or.inl (ne_of_gt <| ZetaSum_aux1_1' apos hx)\n  \u00b7 exact fun x hx h \u21a6 by rw [Real.rpow_eq_zero] at h <;> linarith [ZetaSum_aux1_1' apos hx]\n\nlemma ZetaSum_aux1_5c {a b : \u211d} {s : \u2102} :\n    let g : \u211d \u2192 \u211d := fun u \u21a6 |\u2191\u230au\u230b + 1 / 2 - u| / u ^ (s.re + 1);\n    MeasureTheory.AEStronglyMeasurable g\n      (MeasureTheory.Measure.restrict MeasureTheory.volume (\u0399 a b)) := by\n  intro\n  refine (Measurable.div ?_ <| measurable_id.pow_const _).aestronglyMeasurable\n  refine (_root_.continuous_abs).measurable.comp ?_\n  refine Measurable.sub (Measurable.add ?_ measurable_const) measurable_id\n  exact Measurable.comp (by exact fun _ _ \u21a6 trivial) Int.measurable_floor\n\nlemma ZetaSum_aux1_5d {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} (\u03c3pos : 0 < s.re) :\n  IntervalIntegrable (fun u \u21a6 |\u2191\u230au\u230b + 1 / 2 - u| / u ^ (s.re + 1)) MeasureTheory.volume a b := by\n  set g : \u211d \u2192 \u211d := (fun u \u21a6 |\u2191\u230au\u230b + 1 / 2 - u| / u ^ (s.re + 1))\n  apply IntervalIntegrable.mono_fun (ZetaSum_aux1_5b apos a_lt_b \u03c3pos) ZetaSum_aux1_5c ?_\n  filter_upwards with x\n  simp only [g, Real.norm_eq_abs, one_div, norm_inv, abs_div, _root_.abs_abs]\n  conv => rw [div_eq_mul_inv, \u2190 one_div]; rhs; rw [\u2190 one_mul |x ^ (s.re + 1)|\u207b\u00b9]\n  refine mul_le_mul ?_ (le_refl _) (by simp) <| by norm_num\n  exact le_trans (ZetaSum_aux1_3 x) <| by norm_num\n\n", "theoremStatement": "lemma ZetaSum_aux1_5 {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) 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"Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply intervalIntegral.integral_mono_on (le_of_lt a_lt_b) ?_ ?_\n  \u00b7 exact ZetaSum_aux1_5a apos\n  \u00b7 exact ZetaSum_aux1_5d apos a_lt_b \u03c3pos\n  \u00b7 exact ZetaSum_aux1_5b apos a_lt_b \u03c3pos", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 188}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\nlemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self\n\nlemma bound_sum_log0 {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let f0 i := if i = 0 then 0 else f i\n  have l1 : chebyWith C f0 := by\n    intro n ; refine Finset.sum_le_sum (fun i _ => ?_) |>.trans (hf n)\n    by_cases hi : i = 0 <;> simp [hi, f0]\n  have l2 i : \u2016f i\u2016 / i = \u2016f0 i\u2016 / i := by by_cases hi : i = 0 <;> simp [hi, f0]\n  simp_rw [l2] ; apply bound_sum_log rfl l1 hx\n\nlemma bound_sum_log' {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + 2 * \u03c0 ^ 2) := by\n  simpa only [hh_integral'] using bound_sum_log0 hf hx\n\nlemma summable_fourier (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  exact Summable.of_nonneg_of_le (fun _ => norm_nonneg _) l6 (by simpa using l5.const_smul (W21.norm \u03c8))\n\nlemma bound_I1 (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264\n    W21.norm \u03c8 \u2022 \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 := by\n\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  have l1 : Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n    exact summable_fourier x hx \u03c8 hcheby\n  apply (norm_tsum_le_tsum_norm l1).trans\n  simpa only [\u2190 tsum_const_smul _ l5] using tsum_mono l1 (by simpa using l5.const_smul (W21.norm \u03c8)) l6\n\nlemma bound_I1' {C : \u211d} (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21) (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264 W21.norm \u03c8 * C * (1 + 2 * \u03c0 ^ 2) := by\n\n  apply bound_I1 x (by linarith) \u03c8 \u27e8_, hcheby\u27e9 |>.trans\n  rw [smul_eq_mul, mul_assoc]\n  apply mul_le_mul le_rfl (bound_sum_log' hcheby hx) ?_ W21.norm_nonneg\n  apply tsum_nonneg (fun i => by positivity)\n\nlemma bound_I2 (x : \u211d) (\u03c8 : W21) :\n    \u2016\u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (2 * \u03c0 ^ 2) := by\n\n  have key a : \u2016\ud835\udcd5 \u03c8 (a / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := decay_bounds_key \u03c8 _\n  have twopi : 0 \u2264 2 * \u03c0 := by simp [pi_nonneg]\n  have l3 : Integrable (fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.comp_div (by norm_num [pi_ne_zero])\n  have l2 : IntegrableOn (fun i \u21a6 W21.norm \u03c8 * (1 + (i / (2 * \u03c0)) ^ 2)\u207b\u00b9) (Ici (-Real.log x)) := by\n    exact (l3.const_mul _).integrableOn\n  have l1 : IntegrableOn (fun i \u21a6 \u2016\ud835\udcd5 \u03c8 (i / (2 * \u03c0))\u2016) (Ici (-Real.log x)) := by\n    refine ((l3.const_mul (W21.norm \u03c8)).mono' ?_ ?_).integrableOn\n    \u00b7 apply Continuous.aestronglyMeasurable ; continuity\n    \u00b7 simp only [norm_norm, key] ; simp\n  have l5 : 0 \u2264\u1d50[volume] fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := by apply eventually_of_forall ; intro x ; positivity\n  refine (norm_integral_le_integral_norm _).trans <| (set_integral_mono l1 l2 key).trans ?_\n  rw [integral_mul_left] ; gcongr ; apply W21.norm_nonneg\n  refine (set_integral_le_integral l3 l5).trans ?_\n  rw [Measure.integral_comp_div (fun x => (1 + x ^ 2)\u207b\u00b9) (2 * \u03c0)]\n  simp [abs_eq_self.mpr twopi] ; ring_nf ; rfl\n\nlemma bound_main {C : \u211d} (A : \u2102) (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21)\n    (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264\n      W21.norm \u03c8 * (C * (1 + 2 * \u03c0 ^ 2) + \u2016A\u2016 * (2 * \u03c0 ^ 2)) := by\n\n  have l1 := bound_I1' x hx \u03c8 hcheby\n  have l2 := mul_le_mul (le_refl \u2016A\u2016) (bound_I2 x \u03c8) (by positivity) (by positivity)\n  apply norm_sub_le _ _ |>.trans ; rw [norm_mul]\n  convert _root_.add_le_add l1 l2 using 1 ; ring\n\n/-%%\n\\begin{lemma}[Limiting identity for Schwartz functions]\\label{schwarz-id}\\lean{limiting_cor_schwartz}\\leanok  The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.\n\\end{lemma}\n%%-/\n\nlemma limiting_cor_W21 (\u03c8 : W21) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) := by\n\n  -- Shorter notation for clarity\n  let S1 x (\u03c8 : \u211d \u2192 \u2102) := \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191n / x))\n  let S2 x (\u03c8 : \u211d \u2192 \u2102) := \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\n  let S x \u03c8 := S1 x \u03c8 - S2 x \u03c8 ; change Tendsto (fun x \u21a6 S x \u03c8) atTop (\ud835\udcdd 0)\n\n  -- Build the truncation\n  obtain g := exists_trunc\n  let \u03a8 R := g.scale R * \u03c8\n  have key R : Tendsto (fun x \u21a6 S x (\u03a8 R)) atTop (\ud835\udcdd 0) := limiting_cor (\u03a8 R) hf hcheby hG hG'\n\n  -- Choose the truncation radius\n  obtain \u27e8C, hcheby\u27e9 := hcheby\n  have hC : 0 \u2264 C := by\n    have : \u2016f 0\u2016 \u2264 C := by simpa [cumsum] using hcheby 1\n    have : 0 \u2264 \u2016f 0\u2016 := by positivity\n    linarith\n  have key2 : Tendsto (fun R \u21a6 W21.norm (\u03c8 - \u03a8 R)) atTop (\ud835\udcdd 0) := W21_approximation \u03c8 g\n  simp_rw [Metric.tendsto_nhds] at key key2 \u22a2 ; intro \u03b5 h\u03b5\n  let M := C * (1 + 2 * \u03c0 ^ 2) + \u2016(A : \u2102)\u2016 * (2 * \u03c0 ^ 2)\n  obtain \u27e8R, hR\u03c8\u27e9 := (key2 ((\u03b5 / 2) / (1 + M)) (by positivity)).exists\n  simp only [dist_zero_right, Real.norm_eq_abs, abs_eq_self.mpr W21.norm_nonneg] at hR\u03c8 key\n\n  -- Apply the compact support case\n  filter_upwards [eventually_ge_atTop 1, key R (\u03b5 / 2) (by positivity)] with x hx key\n\n  -- Control the tail term\n  have key3 : \u2016S x (\u03c8 - \u03a8 R)\u2016 < \u03b5 / 2 := by\n    have : \u2016S x _\u2016 \u2264 _ * M := @bound_main f C A x hx (\u03c8 - \u03a8 R) hcheby\n    apply this.trans_lt\n    apply (mul_le_mul (d := 1 + M) le_rfl (by simp) (by positivity) W21.norm_nonneg).trans_lt\n    have : 0 < 1 + M := by positivity\n    convert (mul_lt_mul_right this).mpr hR\u03c8 using 1 ; field_simp ; ring\n\n  -- Conclude the proof\n  have S1_sub_1 x : \ud835\udcd5 (\u21d1\u03c8 - \u21d1(\u03a8 R)) x = \ud835\udcd5 \u03c8 x - \ud835\udcd5 (\u03a8 R) x := by\n    have l1 : AEStronglyMeasurable (fun x_1 : \u211d \u21a6 cexp (-(2 * \u2191\u03c0 * (\u2191x_1 * \u2191x) * I))) volume := by\n      refine (Continuous.mul ?_ continuous_const).neg.cexp.aestronglyMeasurable\n      apply continuous_const.mul <| contDiff_ofReal.continuous.mul continuous_const\n    simp [Real.fourierIntegral_eq', mul_sub] ; apply integral_sub\n    \u00b7 apply \u03c8.hf.bdd_mul l1 ; use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n    \u00b7 apply (\u03a8 R : W21) |>.hf |>.bdd_mul l1\n      use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n\n  have S1_sub : S1 x (\u03c8 - \u03a8 R) = S1 x \u03c8 - S1 x (\u03a8 R) := by\n    simp [S1, S1_sub_1, mul_sub] ; apply tsum_sub\n    \u00b7 have := summable_fourier x (by positivity) \u03c8 \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n    \u00b7 have := summable_fourier x (by positivity) (\u03a8 R) \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n\n  have S2_sub : S2 x (\u03c8 - \u03a8 R) = S2 x \u03c8 - S2 x (\u03a8 R) := by\n    simp [S2, S1_sub_1] ; rw [integral_sub] ; ring\n    \u00b7 exact \u03c8.integrable_fourier (by positivity) |>.restrict\n    \u00b7 exact (\u03a8 R : W21).integrable_fourier (by positivity) |>.restrict\n\n  have S_sub : S x (\u03c8 - \u03a8 R) = S x \u03c8 - S x (\u03a8 R) := by simp [S, S1_sub, S2_sub] ; ring\n  simpa [S_sub, \u03a8] using norm_add_le _ _ |>.trans_lt (_root_.add_lt_add key3 key)\n\nlemma limiting_cor_schwartz (\u03c8 : \ud835\udce2(\u211d, \u2102)) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) :=\n  limiting_cor_W21 \u03c8 hf hcheby hG hG'\n\n/-%%\n\\begin{proof}\n\\uses{limiting-cor, smooth-ury}\\leanok\nFor any $R>1$, one can use a smooth cutoff function (provided by Lemma \\ref{smooth-ury} to write $\\psi = \\psi_{\\leq R} + \\psi_{>R}$, where $\\psi_{\\leq R}$ is $C^2$ (in fact smooth) and compactly supported (on $[-R,R]$), and $\\psi_{>R}$ obeys bounds of the form\n$$ |\\psi_{>R}(t)|, |\\psi''_{>R}(t)| \\ll R^{-1} / (1 + |t|^2) $$\nwhere the implied constants depend on $\\psi$.  By Lemma \\ref{decay} we then have\n$$ \\hat \\psi_{>R}(u) \\ll R^{-1} / (1+|u|^2).$$\nUsing this and \\eqref{cheby} one can show that\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{>R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ), A \\int_{-\\infty}^\\infty \\hat \\psi_{>R} (\\frac{u}{2\\pi})\\ du \\ll R^{-1} $$\n(with implied constants also depending on $A$), while from Lemma \\ref{limiting-cor} one has\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{\\leq R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi_{\\leq R} (\\frac{u}{2\\pi})\\ du + o(1).$$\nCombining the two estimates and letting $R$ be large, we obtain the claim.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Bijectivity of Fourier transform]\\label{bij}\\lean{fourier_surjection_on_schwartz}\\leanok  The Fourier transform is a bijection on the Schwartz class.\n\\end{lemma}\n%%-/\n\n-- just the surjectivity is stated here, as this is all that is needed for the current application, but perhaps one should state and prove bijectivity instead\n\nlemma fourier_surjection_on_schwartz (f : \ud835\udce2(\u211d, \u2102)) : \u2203 g : \ud835\udce2(\u211d, \u2102), \ud835\udcd5 g = f := by\n  use FS (FS (FS f)) ; ext x ; nth_rewrite 2 [\u2190 FS4 f] ; simp\n\n/-%%\n\\begin{proof}\n  \\leanok\n This is a standard result in Fourier analysis.\nIt can be proved here by appealing to Mellin inversion, Theorem \\ref{MellinInversion}.\nIn particular, given $f$ in the Schwartz class, let $F : \\R_+ \\to \\C : x \\mapsto f(\\log x)$ be a function in the ``Mellin space''; then the Mellin transform of $F$ on the imaginary axis $s=it$ is the Fourier transform of $f$.  The Mellin inversion theorem gives Fourier inversion.\n\\end{proof}\n%%-/\n\ndef toSchwartz (f : \u211d \u2192 \u2102) (h1 : ContDiff \u211d \u22a4 f) (h2 : HasCompactSupport f) : \ud835\udce2(\u211d, \u2102) where\n  toFun := f\n  smooth' := h1\n  decay' k n := by\n    have l1 : Continuous (fun x => \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := by\n      have : ContDiff \u211d \u22a4 (iteratedFDeriv \u211d n f) := h1.iteratedFDeriv_right le_top\n      exact Continuous.mul (by continuity) this.continuous.norm\n    have l2 : HasCompactSupport (fun x \u21a6 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := (h2.iteratedFDeriv _).norm.mul_left\n    simpa using l1.bounded_above_of_compact_support l2\n\n@[simp] lemma toSchwartz_apply (f : \u211d \u2192 \u2102) {h1 h2 x} : SchwartzMap.mk f h1 h2 x = f x := rfl\n\nlemma comp_exp_support0 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in \ud835\udcdd 0, \u03a8 x = 0 :=\n  not_mem_tsupport_iff_eventuallyEq.mp (fun h => lt_irrefl 0 <| mem_Ioi.mp (hplus h))\n\nlemma comp_exp_support1 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in atBot, \u03a8 (exp x) = 0 :=\n  Real.tendsto_exp_atBot <| comp_exp_support0 hplus\n\nlemma comp_exp_support2 {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) :\n    \u2200\u1da0 (x : \u211d) in atTop, (\u03a8 \u2218 rexp) x = 0 := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop] at hsupp\n  exact Real.tendsto_exp_atTop hsupp.2\n\ntheorem comp_exp_support {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    HasCompactSupport (\u03a8 \u2218 rexp) := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop]\n  exact \u27e8comp_exp_support1 hplus, comp_exp_support2 hsupp\u27e9\n\nlemma wiener_ikehara_smooth_aux (l0 : Continuous \u03a8) (hsupp : HasCompactSupport \u03a8)\n    (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) (x : \u211d) (hx : 0 < x) :\n    \u222b (u : \u211d) in Ioi (-Real.log x), \u2191(rexp u) * \u03a8 (rexp u) = \u222b (y : \u211d) in Ioi (1 / x), \u03a8 y := by\n\n  have l1 : ContinuousOn rexp (Ici (-Real.log x)) := by fun_prop\n  have l2 : Tendsto rexp atTop atTop := Real.tendsto_exp_atTop\n  have l3 t (_ : t \u2208 Ioi (-log x)) : HasDerivWithinAt rexp (rexp t) (Ioi t) t :=\n    (Real.hasDerivAt_exp t).hasDerivWithinAt\n  have l4 : ContinuousOn \u03a8 (rexp '' Ioi (-Real.log x)) := by fun_prop\n  have l5 : IntegrableOn \u03a8 (rexp '' Ici (-Real.log x)) volume :=\n    (l0.integrable_of_hasCompactSupport hsupp).integrableOn\n  have l6 : IntegrableOn (fun x \u21a6 rexp x \u2022 (\u03a8 \u2218 rexp) x) (Ici (-Real.log x)) volume := by\n    refine (Continuous.integrable_of_hasCompactSupport (by continuity) ?_).integrableOn\n    change HasCompactSupport (rexp \u2022 (\u03a8 \u2218 rexp))\n    exact (comp_exp_support hsupp hplus).smul_left\n  have := MeasureTheory.integral_comp_smul_deriv_Ioi l1 l2 l3 l4 l5 l6\n  simpa [Real.exp_neg, Real.exp_log hx] using this\n\ntheorem wiener_ikehara_smooth_sub (h1 : Integrable \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    Tendsto (fun x \u21a6 (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n\n  obtain \u27e8\u03b5, h\u03b5, hh\u27e9 := Metric.eventually_nhds_iff.mp <| comp_exp_support0 hplus\n  apply tendsto_nhds_of_eventually_eq ; filter_upwards [eventually_gt_atTop \u03b5\u207b\u00b9] with x hx\u03b5\n\n  have l1 : Integrable (indicator (Ioi x\u207b\u00b9) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n  have l2 : Integrable (indicator (Ioi 0) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n\n  simp_rw [\u2190 MeasureTheory.integral_indicator measurableSet_Ioi, \u2190 mul_sub, \u2190 integral_sub l1 l2]\n  simp ; right ; apply MeasureTheory.integral_eq_zero_of_ae ; apply eventually_of_forall ; intro t ; simp\n\n  have h\u03b5' : 0 < \u03b5\u207b\u00b9 := by positivity\n  have hx : 0 < x := by linarith\n  have hx' : 0 < x\u207b\u00b9 := by positivity\n  have h\u03b5x : x\u207b\u00b9 < \u03b5 := by apply (inv_lt h\u03b5 hx).mp hx\u03b5\n\n  have l3 : Ioi 0 = Ioc 0 x\u207b\u00b9 \u222a Ioi x\u207b\u00b9 := by\n    ext t ; simp ; constructor <;> intro h\n    \u00b7 simp [h, le_or_lt]\n    \u00b7 cases h <;> linarith\n  have l4 : Disjoint (Ioc 0 x\u207b\u00b9) (Ioi x\u207b\u00b9) := by simp\n  have l5 := Set.indicator_union_of_disjoint l4 \u03a8\n  rw [l3, l5] ; ring_nf\n  by_cases ht : t \u2208 Ioc 0 x\u207b\u00b9 <;> simp [ht]\n  apply hh ; simp at ht \u22a2\n  have : |t| \u2264 x\u207b\u00b9 := by rw [abs_le] ; constructor <;> linarith\n  linarith\n\n/-%%\n\\begin{corollary}[Smoothed Wiener-Ikehara]\\label{WienerIkeharaSmooth}\\lean{wiener_ikehara_smooth}\\leanok\n  If $\\Psi: (0,\\infty) \\to \\C$ is smooth and compactly supported away from the origin, then, then\n$$ \\sum_{n=1}^\\infty f(n) \\Psi( \\frac{n}{x} ) = A x \\int_0^\\infty \\Psi(y)\\ dy + o(x)$$\nas $u \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma wiener_ikehara_smooth (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x - A * \u222b y in Set.Ioi 0, \u03a8 y) atTop (nhds 0) := by\n\n  let h (x : \u211d) : \u2102 := rexp (2 * \u03c0 * x) * \u03a8 (exp (2 * \u03c0 * x))\n  have h1 : ContDiff \u211d \u22a4 h := by\n    have : ContDiff \u211d \u22a4 (fun x : \u211d => (rexp (2 * \u03c0 * x))) := (contDiff_const.mul contDiff_id).exp\n    exact (contDiff_ofReal.comp this).mul (hsmooth.comp this)\n  have h2 : HasCompactSupport h := by\n    have : 2 * \u03c0 \u2260 0 := by simp [pi_ne_zero]\n    simpa using (comp_exp_support hsupp hplus).comp_smul this |>.mul_left\n  obtain \u27e8g, hg\u27e9 := fourier_surjection_on_schwartz (toSchwartz h h1 h2)\n\n  have why (x : \u211d) : 2 * \u03c0 * x / (2 * \u03c0) = x := by field_simp ; ring\n  have l1 {y} (hy : 0 < y) : y * \u03a8 y = \ud835\udcd5 g (1 / (2 * \u03c0) * Real.log y) := by\n    field_simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [Real.exp_log hy]\n\n  have key := limiting_cor_schwartz g hf hcheby hG hG'\n\n  have l2 : \u2200\u1da0 x in atTop, \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 (\u21d1g) (1 / (2 * \u03c0) * Real.log (\u2191n / x)) =\n      \u2211' (n : \u2115), f n * \u03a8 (\u2191n / x) / x := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr ; ext n\n    by_cases hn : n = 0 ; simp [hn, (comp_exp_support0 hplus).self_of_nhds]\n    rw [\u2190 l1 (by positivity)]\n    have : (n : \u2102) \u2260 0 := by simpa using hn\n    have : (x : \u2102) \u2260 0 := by simpa using hx.ne.symm\n    field_simp ; ring\n\n  have l3 : \u2200\u1da0 x in atTop, \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 (\u21d1g) (u / (2 * \u03c0)) =\n      \u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr 1 ; simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [MeasureTheory.integral_Ici_eq_integral_Ioi]\n    exact wiener_ikehara_smooth_aux hsmooth.continuous hsupp hplus x hx\n\n  have l4 : Tendsto (fun x => (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n    exact wiener_ikehara_smooth_sub (hsmooth.continuous.integrable_of_hasCompactSupport hsupp) hplus\n\n  simpa [tsum_div_const] using (key.congr' <| EventuallyEq.sub l2 l3) |>.add l4\n\n/-%%\n\\begin{proof}\n\\uses{bij,schwarz-id}\\leanok\n By Lemma \\ref{bij}, we can write\n$$ y \\Psi(y) = \\hat \\psi( \\frac{1}{2\\pi} \\log y )$$\nfor all $y>0$ and some Schwartz function $\\psi$.  Making this substitution, the claim is then equivalent after standard manipulations to\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\nand the claim follows from Lemma \\ref{schwarz-id}.\n\\end{proof}\n%%-/\n\nlemma wiener_ikehara_smooth' (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) :=\n  tendsto_sub_nhds_zero_iff.mp <| wiener_ikehara_smooth hf hcheby hG hG' hsmooth hsupp hplus\n\nlocal instance {E : Type*} : Coe (E \u2192 \u211d) (E \u2192 \u2102) := \u27e8fun f n => f n\u27e9\n\n@[norm_cast]\ntheorem set_integral_ofReal {f : \u211d \u2192 \u211d} {s : Set \u211d} : \u222b x in s, (f x : \u2102) = \u222b x in s, f x :=\n  integral_ofReal\n\nlemma wiener_ikehara_smooth_real {f : \u2115 \u2192 \u211d} {\u03a8 : \u211d \u2192 \u211d} (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) := by\n\n  let \u03a8' := ofReal' \u2218 \u03a8\n  have l1 : ContDiff \u211d \u22a4 \u03a8' := contDiff_ofReal.comp hsmooth\n  have l2 : HasCompactSupport \u03a8' := hsupp.comp_left rfl\n  have l3 : closure (Function.support \u03a8') \u2286 Ioi 0 := by rwa [Function.support_comp_eq] ; simp\n  have key := (continuous_re.tendsto _).comp (@wiener_ikehara_smooth' A \u03a8 G f hf hcheby hG hG' l1 l2 l3)\n  simp at key ; norm_cast at key\n\nlemma interval_approx_inf (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      \u03c8 \u2264 indicator (Ico a b) 1 \u2227 b - a - \u03b5 \u2264 \u222b y in Ioi 0, \u03c8 y := by\n\n  have l1 : Iio ((b - a) / 3) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < (b - a) / 3)\n  have l2 : a < a + \u03b5 / 2 := by linarith\n  have l3 : b - \u03b5 / 2 < b := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 simp [h5, hab.ne, Icc_subset_Ioi_iff hab.le, ha]\n  \u00b7 exact h4.trans <| indicator_le_indicator_of_subset Ioo_subset_Ico_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a - \u03b5 := by linarith\n    have l5 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2229 Ioi 0 = Icc (a + \u03b5 / 2) (b - \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Icc (a + \u03b5 / 2) (b - \u03b5 / 2)) 1 y = b - a - \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; apply set_integral_mono ?_ l8 h3\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Icc]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma interval_approx_sup (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      indicator (Ico a b) 1 \u2264 \u03c8 \u2227 \u222b y in Ioi 0, \u03c8 y \u2264 b - a + \u03b5 := by\n\n  have l1 : Iio (a / 2) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < a / 2)\n  have l2 : a - \u03b5 / 2 < a := by linarith\n  have l3 : b < b + \u03b5 / 2 := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 have l4 : a - \u03b5 / 2 < b + \u03b5 / 2 := by linarith\n    have l5 : \u03b5 / 2 < a := by linarith\n    simp [h5, l4.ne, Icc_subset_Ioi_iff l4.le, l5]\n  \u00b7 apply le_trans ?_ h3\n    apply indicator_le_indicator_of_subset Ico_subset_Icc_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a + \u03b5 := by linarith\n    have l5 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2229 Ioi 0 = Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Ioo (a - \u03b5 / 2) (b + \u03b5 / 2)) 1 y = b - a + \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; refine set_integral_mono l8 ?_ h4\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Ioo]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma WI_summable {f : \u2115 \u2192 \u211d} {g : \u211d \u2192 \u211d} (hg : HasCompactSupport g) (hx : 0 < x) :\n    Summable (fun n => f n * g (n / x)) := by\n  obtain \u27e8M, hM\u27e9 := hg.bddAbove.mono subset_closure\n  apply summable_of_finite_support\n  simp ; apply Finite.inter_of_right ; rw [finite_iff_bddAbove]\n  exact \u27e8Nat.ceil (M * x), fun i hi => by simpa using Nat.ceil_mono ((div_le_iff hx).mp (hM hi))\u27e9\n\nlemma WI_sum_le {f : \u2115 \u2192 \u211d} {g\u2081 g\u2082 : \u211d \u2192 \u211d} (hf : 0 \u2264 f) (hg : g\u2081 \u2264 g\u2082) (hx : 0 < x)\n    (hg\u2081 : HasCompactSupport g\u2081) (hg\u2082 : HasCompactSupport g\u2082) :\n    (\u2211' n, f n * g\u2081 (n / x)) / x \u2264 (\u2211' n, f n * g\u2082 (n / x)) / x := by\n  apply div_le_div_of_nonneg_right ?_ hx.le\n  exact tsum_le_tsum (fun n => mul_le_mul_of_nonneg_left (hg _) (hf _)) (WI_summable hg\u2081 hx) (WI_summable hg\u2082 hx)\n\nlemma WI_sum_Iab_le {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) {C : \u211d} (hcheby : chebyWith C f) (hb : 0 < b) (hxb : 2 / b < x) :\n    (\u2211' n, f n * indicator (Ico a b) 1 (n / x)) / x \u2264 C * 2 * b := by\n  have hb' : 0 < 2 / b := by positivity\n  have hx : 0 < x := by linarith\n  have hxb' : 2 < x * b := (div_lt_iff hb).mp hxb\n  have l1 (i : \u2115) (hi : i \u2209 Finset.range \u2308b * x\u2309\u208a) : f i * indicator (Ico a b) 1 (i / x) = 0 := by\n    simp at hi \u22a2 ; right ; rintro - ; rw [le_div_iff hx] ; linarith\n  have l2 (i : \u2115) (_ : i \u2208 Finset.range \u2308b * x\u2309\u208a) : f i * indicator (Ico a b) 1 (i / x) \u2264 |f i| := by\n    rw [abs_eq_self.mpr (hpos _)]\n    convert_to _ \u2264 f i * 1 ; ring\n    apply mul_le_mul_of_nonneg_left ?_ (hpos _)\n    by_cases hi : (i / x) \u2208 (Ico a b) <;> simp [hi]\n  rw [tsum_eq_sum l1, div_le_iff hx, mul_assoc, mul_assoc]\n  apply Finset.sum_le_sum l2 |>.trans\n  have := hcheby \u2308b * x\u2309\u208a ; simp at this ; apply this.trans\n  have : 0 \u2264 C := by have := hcheby 1 ; simp [cumsum] at this ; exact (abs_nonneg _).trans this\n  refine mul_le_mul_of_nonneg_left ?_ this\n  apply (Nat.ceil_lt_add_one (by positivity)).le.trans\n  linarith\n\nlemma WI_sum_Iab_le' {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) {C : \u211d} (hcheby : chebyWith C f) (hb : 0 < b) :\n    \u2200\u1da0 x : \u211d in atTop, (\u2211' n, f n * indicator (Ico a b) 1 (n / x)) / x \u2264 C * 2 * b := by\n  filter_upwards [eventually_gt_atTop (2 / b)] with x hx using WI_sum_Iab_le hpos hcheby hb hx\n\nlemma le_of_eventually_nhdsWithin {a b : \u211d} (h : \u2200\u1da0 c in \ud835\udcdd[>] b, a \u2264 c) : a \u2264 b := by\n  apply le_of_forall_lt' ; intro d hd\n  have key : \u2200\u1da0 c in \ud835\udcdd[>] b, c < d := by\n    apply eventually_of_mem (U := Iio d) ?_ (fun x hx => hx)\n    rw [mem_nhdsWithin]\n    refine \u27e8Iio d, isOpen_Iio, hd, inter_subset_left _ _\u27e9\n  obtain \u27e8x, h1, h2\u27e9 := (h.and key).exists\n  linarith\n\nlemma ge_of_eventually_nhdsWithin {a b : \u211d} (h : \u2200\u1da0 c in \ud835\udcdd[<] b, c \u2264 a) : b \u2264 a := by\n  apply le_of_forall_lt ; intro d hd\n  have key : \u2200\u1da0 c in \ud835\udcdd[<] b, c > d := by\n    apply eventually_of_mem (U := Ioi d) ?_ (fun x hx => hx)\n    rw [mem_nhdsWithin]\n    refine \u27e8Ioi d, isOpen_Ioi, hd, inter_subset_left _ _\u27e9\n  obtain \u27e8x, h1, h2\u27e9 := (h.and key).exists\n  linarith\n\nlemma WI_tendsto_aux (a b : \u211d) {A : \u211d} (hA : 0 < A) :\n    Tendsto (fun c => c / A - (b - a)) (\ud835\udcdd[>] (A * (b - a))) (\ud835\udcdd[>] 0) := by\n  rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n  intro \u03b5 h\u03b5\n  refine \u27e8A * \u03b5, by positivity, ?_\u27e9\n  intro x hx1 hx2\n  constructor\n  \u00b7 simpa [lt_div_iff' hA]\n  \u00b7 simp [Real.dist_eq] at hx2 \u22a2\n    have : |x / A - (b - a)| = |x - A * (b - a)| / A := by\n      rw [\u2190 abs_eq_self.mpr hA.le, \u2190 abs_div, abs_eq_self.mpr hA.le] ; congr ; field_simp\n    rwa [this, div_lt_iff' hA]\n\nlemma WI_tendsto_aux' (a b : \u211d) {A : \u211d} (hA : 0 < A) :\n    Tendsto (fun c => (b - a) - c / A) (\ud835\udcdd[<] (A * (b - a))) (\ud835\udcdd[>] 0) := by\n  rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n  intro \u03b5 h\u03b5\n  refine \u27e8A * \u03b5, by positivity, ?_\u27e9\n  intro x hx1 hx2\n  constructor\n  \u00b7 simpa [div_lt_iff' hA]\n  \u00b7 simp [Real.dist_eq] at hx2 \u22a2\n    have : |(b - a) - x / A| = |A * (b - a) - x| / A := by\n      rw [\u2190 abs_eq_self.mpr hA.le, \u2190 abs_div, abs_eq_self.mpr hA.le] ; congr ; field_simp ; ring\n    rwa [this, div_lt_iff' hA, \u2190 neg_sub, abs_neg]\n\n", "theoremStatement": "theorem residue_nonneg {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f)\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm (fun n \u21a6 \u2191(f n)) \u03c3')) (hcheby : cheby fun n \u21a6 \u2191(f n))\n    (hG : ContinuousOn G {s | 1 \u2264 s.re}) (hG' : EqOn G (fun s \u21a6 LSeries (fun n \u21a6 \u2191(f n)) s - \u2191A / (s - 1)) {s | 1 < s.re}) : 0 \u2264 A ", "theoremName": "residue_nonneg", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "dea220be97c74ee2bce9acc75cd14ec7bc75afc4", "date": "2024-04-16"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1907, "tokenPositionInFile": 95584, "theoremPositionInFile": 152}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 349, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", 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"Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let S (g : \u211d \u2192 \u211d) (x : \u211d) := (\u2211' n, f n * g (n / x)) / x\n  have hSnonneg {g : \u211d \u2192 \u211d} (hg : 0 \u2264 g) : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S g x := by\n    filter_upwards [eventually_ge_atTop 0] with x hx\n    exact div_nonneg (tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))) hx\n  obtain \u27e8\u03b5, \u03c8, h1, h2, h3, h4, -\u27e9 := (interval_approx_sup zero_lt_one one_lt_two).exists\n  have key := @wiener_ikehara_smooth_real A G f \u03c8 hf hcheby hG hG' h1 h2 h3\n  have l2 : 0 \u2264 \u03c8 := by apply le_trans _ h4 ; apply indicator_nonneg ; simp\n  have l1 : \u2200\u1da0 x in atTop, 0 \u2264 S \u03c8 x := hSnonneg l2\n  have l3 : 0 \u2264 A * \u222b (y : \u211d) in Ioi 0, \u03c8 y := ge_of_tendsto key l1\n  have l4 : 0 < \u222b (y : \u211d) in Ioi 0, \u03c8 y := by\n    have r1 : 0 \u2264\u1d50[Measure.restrict volume (Ioi 0)] \u03c8 := eventually_of_forall l2\n    have r2 : IntegrableOn (fun y \u21a6 \u03c8 y) (Ioi 0) volume :=\n      (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    have r3 : Ico 1 2 \u2286 Function.support \u03c8 := by intro x hx ; have := h4 x ; simp [hx] at this \u22a2 ; linarith\n    have r4 : Ico 1 2 \u2286 Function.support \u03c8 \u2229 Ioi 0 := by\n      simp [r3] ; apply Ico_subset_Icc_self.trans ; rw [Icc_subset_Ioi_iff] <;> linarith\n    have r5 : 1 \u2264 volume ((Function.support fun y \u21a6 \u03c8 y) \u2229 Ioi 0) := by convert volume.mono r4 ; norm_num\n    simpa [set_integral_pos_iff_support_of_nonneg_ae r1 r2] using zero_lt_one.trans_le r5\n  have := div_nonneg l3 l4.le ; field_simp at this ; exact this", "proofType": "tactic", "proofLengthLines": 19, "proofLengthTokens": 1403}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]\n\nlemma Filter.BigO_zero_atTop_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[atTop] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioi_mem_atTop b] with c hc; replace hc := mem_Ioi.mp hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h]\n\n-- steal coerction lemmas from EulerProducts.Auxiliary because of build issues, and add new ones\nnamespace Complex\n-- see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Differentiability.20of.20the.20natural.20map.20.E2.84.9D.20.E2.86.92.20.E2.84.82/near/418095234\n\nlemma hasDerivAt_ofReal (x : \u211d) : HasDerivAt ofReal' 1 x :=\n  HasDerivAt.ofReal_comp <| hasDerivAt_id x\n\nlemma deriv_ofReal (x : \u211d) : deriv ofReal' x = 1 :=\n  (hasDerivAt_ofReal x).deriv\n\nlemma differentiableAt_ofReal (x : \u211d) : DifferentiableAt \u211d ofReal' x :=\n  (hasDerivAt_ofReal x).differentiableAt\n\nlemma differentiable_ofReal : Differentiable \u211d ofReal' :=\n  ofRealCLM.differentiable\n\nend Complex\n\nlemma DifferentiableAt.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    DifferentiableAt \u211d (fun x : \u211d \u21a6 e x) z :=\n  hf.hasDerivAt.comp_ofReal.differentiableAt\n\nlemma Differentiable.comp_ofReal {e : \u2102 \u2192 \u2102} (h : Differentiable \u2102 e) :\n    Differentiable \u211d (fun x : \u211d \u21a6 e x) :=\n  fun _ \u21a6 h.differentiableAt.comp_ofReal\n\nlemma DifferentiableAt.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} (hf : DifferentiableAt \u211d f z) :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z :=\n  hf.hasDerivAt.ofReal_comp.differentiableAt\n\nlemma Differentiable.ofReal_comp {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) :=\n  fun _ \u21a6 hf.differentiableAt.ofReal_comp\n\nopen Complex ContinuousLinearMap in\nlemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} {u : \u2102}\n    (hf : HasDerivAt (fun y \u21a6 (f y : \u2102)) u z) :\n    \u2203 u' : \u211d, u = u' \u2227 HasDerivAt f u' z := by\n  lift u to \u211d\n  \u00b7 have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv\n    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H\n    rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H\n  refine \u27e8u, rfl, ?_\u27e9\n  convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt\n  rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]\n\nlemma DifferentiableAt.ofReal_comp_iff {z : \u211d} {f : \u211d \u2192 \u211d} :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z \u2194 DifferentiableAt \u211d f z := by\n  refine \u27e8fun H \u21a6 ?_, ofReal_comp\u27e9\n  obtain \u27e8u, _, hu\u2082\u27e9 := H.hasDerivAt.of_hasDerivAt_ofReal_comp\n  exact HasDerivAt.differentiableAt hu\u2082\n\nlemma Differentiable.ofReal_comp_iff {f : \u211d \u2192 \u211d} :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) \u2194 Differentiable \u211d f :=\n  forall_congr' fun _ \u21a6 DifferentiableAt.ofReal_comp_iff\n\nlemma deriv.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} :\n    deriv (fun (y : \u211d) \u21a6 (f y : \u2102)) z = deriv f z := by\n  by_cases hf : DifferentiableAt \u211d f z\n  \u00b7 exact hf.hasDerivAt.ofReal_comp.deriv\n  \u00b7 have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf\n    rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt <| hf',\n      Complex.ofReal_zero]\n\nlemma deriv.ofReal_comp' {f : \u211d \u2192 \u211d} :\n    deriv (fun x : \u211d \u21a6 (f x : \u2102)) = (fun x \u21a6 ((deriv f) x : \u2102)) :=\n  funext fun _ \u21a6 deriv.ofReal_comp\n\nlemma deriv.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    deriv (fun x : \u211d \u21a6 e x) z = deriv e z :=\n  hf.hasDerivAt.comp_ofReal.deriv\n\nlemma deriv.comp_ofReal' {e : \u2102 \u2192 \u2102} (hf : Differentiable \u2102 e) :\n    deriv (fun x : \u211d \u21a6 e x) = fun (x : \u211d) \u21a6 deriv e x :=\n  funext fun _ \u21a6 deriv.comp_ofReal (hf.differentiableAt)\n\n/-%%\n\\begin{lemma}[PartialIntegration]\\label{PartialIntegration}\\lean{PartialIntegration}\\leanok\nLet $f, g$ be once differentiable functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ so that $fg'$\nand $f'g$ are both integrable, and $f*g (x)\\to 0$ as $x\\to 0^+,\\infty$.\nThen\n$$\n\\int_0^\\infty f(x)g'(x) dx = -\\int_0^\\infty f'(x)g(x)dx.\n$$\n\\end{lemma}\n%%-/\n/-- *Need differentiability, and decay at `0` and `\u221e`* -/\nlemma PartialIntegration (f g : \u211d \u2192 \u2102)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0))\n    (gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0))\n    (lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  simpa using integral_Ioi_mul_deriv_eq_deriv_mul\n    (fun x hx \u21a6 fDiff.hasDerivAt (Ioi_mem_nhds hx))\n    (fun x hx \u21a6 gDiff.hasDerivAt (Ioi_mem_nhds hx))\n    fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\nlemma PartialIntegration_of_support_in_Icc {a b : \u211d} (f g : \u211d \u2192 \u2102) (ha : 0 < a) (h : a \u2264 b)\n    (fSupp : f.support \u2286 Set.Icc a b)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fderivCont : ContinuousOn (deriv f) (Ioi 0))\n    (gderivCont : ContinuousOn (deriv g) (Ioi 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  have Icc_sub : Icc a b \u2286 Ioi 0 := (Icc_subset_Ioi_iff h).mpr ha\n  have fderivSupp := Function.support_deriv_subset_Icc fSupp\n  have fgSupp : (f * g).support \u2286 Icc a b := Function.support_mul_subset_of_subset fSupp\n  have fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fDiff.continuousOn.mono Icc_sub\n    \u00b7 exact gderivCont.mono Icc_sub\n  have gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fderivSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fderivCont.mono Icc_sub\n    \u00b7 exact gDiff.continuousOn.mono Icc_sub\n  have lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0) := TendstoAtZero_of_support_in_Icc (f * g) ha fgSupp\n  have lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0) := TendstoAtTop_of_support_in_Icc (f * g) fgSupp\n  apply PartialIntegration f g fDiff gDiff fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n\n/-%%\nIn this section, we define the Mellin transform (already in Mathlib, thanks to David Loeffler),\nprove its inversion formula, and\nderive a number of important properties of some special functions and bumpfunctions.\n\nDef: (Already in Mathlib)\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be the function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\n[Note: My preferred way to think about this is that we are integrating over the multiplicative\ngroup $\\mathbb{R}_{>0}$, multiplying by a (not necessarily unitary!) character $|\\cdot|^s$, and\nintegrating with respect to the invariant Haar measure $dx/x$. This is very useful in the kinds\nof calculations carried out below. But may be more difficult to formalize as things now stand. So\nwe might have clunkier calculations, which ``magically'' turn out just right - of course they're\nexplained by the aforementioned structure...]\n\n%%-/\n\n\n/-%%\n\\begin{definition}[MellinTransform]\\label{MellinTransform}\\lean{MellinTransform}\\leanok\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be\nthe function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\\end{definition}\n[Note: already exists in Mathlib, with some good API.]\n%%-/\nnoncomputable def MellinTransform (f : \u211d \u2192 \u2102) (s : \u2102) : \u2102 :=\n  \u222b x in Ioi 0, f x * x ^ (s - 1)\n\nlocal notation (name := mellintransform) \"\ud835\udcdc\" => MellinTransform\n/-%%\n\\begin{definition}[MellinInverseTransform]\\label{MellinInverseTransform}\n\\lean{MellinInverseTransform}\\leanok\nLet $F$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$. We define the Mellin inverse transform of\n$F$ to be the function $\\mathcal{M}^{-1}(F)$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}^{-1}(F)(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}F(s)x^{-s}ds,$$\nwhere $\\sigma$ is sufficiently large (say $\\sigma>2$).\n\\end{definition}\n%%-/\nnoncomputable def MellinInverseTransform (F : \u2102 \u2192 \u2102) (\u03c3 : \u211d) (x : \u211d) : \u2102 :=\n  VerticalIntegral' (fun s \u21a6 x ^ (-s) * F s) \u03c3\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_lt]\\label{PerronInverseMellin_lt}\\lean{PerronInverseMellin_lt}\n\\leanok\nLet $0 < t < x$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function\n$$F: s\\mapsto t^s/s(s+1)$$ is equal to\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds\n= 0.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_lt {t x : \u211d} (tpos : 0 < t) (t_lt_x : t < x) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 0 := by\n  dsimp [MellinInverseTransform, VerticalIntegral']\n  have xpos : 0 < x := lt_trans tpos t_lt_x\n  simp only [one_div, mul_inv_rev, inv_I, neg_mul, neg_eq_zero, mul_eq_zero, I_ne_zero,\n    inv_eq_zero, ofReal_eq_zero, pi_ne_zero, OfNat.ofNat_ne_zero, or_self, false_or]\n  convert Perron.formulaLtOne (div_pos tpos xpos) ((div_lt_one xpos).mpr t_lt_x) \u03c3pos using 2\n  ext1 s\n  convert Perron.f_mul_eq_f tpos xpos s using 1\n  ring\n/-%%\n\\begin{proof}\\leanok\n\\uses{Perron.formulaLtOne}\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_gt]\\label{PerronInverseMellin_gt}\\lean{PerronInverseMellin_gt}\n\\leanok\nLet $0 < x < t$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function is equal\nto\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds = 1 - x / t.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_gt {t x : \u211d} (xpos : 0 < x) (x_lt_t : x < t) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 1 - x / t := by\n  dsimp [MellinInverseTransform]\n  have tpos : 0 < t := by linarith\n  have txinv_gtOne : 1 < t / x := (one_lt_div xpos).mpr x_lt_t\n  rw [\u2190 smul_eq_mul]\n  convert Perron.formulaGtOne txinv_gtOne \u03c3pos using 2\n  \u00b7 congr\n    ext1 s\n    convert Perron.f_mul_eq_f tpos xpos s using 1\n    ring\n  \u00b7 field_simp\n/-%%\n\\begin{proof}\n\\uses{Perron.formulaGtOne}\\leanok\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-% ** Wrong delimiters on purpose **\nUnnecessary lemma:\n%\\begin{lemma}[MellinInversion_aux1]\\label{MellinInversion_aux1}\\lean{MellinInversion_aux1}\\leanok\nLet $f$ be differentiable on $(0,\\infty)$, and assume that $f(x)x^s\\to 0$ as $x\\to 0$, and that\n$f(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f(x)x^s\\frac{dx}{x} = \\frac{1}{s}\\int_0^\\infty f'(x)x^{s} dx.\n$$\n%\\end{lemma}\n%-/\nlemma MellinInversion_aux1 {f : \u211d \u2192 \u2102} {s : \u2102} (s_ne_zero : s \u2260 0)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * x ^ s / x = - \u222b x in Ioi 0, (deriv f x) * x ^ s / s := by\n  sorry\n\n/-% ** Wrong delimiters on purpose **\n\\begin{proof}\n\\uses{PartialIntegration}\nPartial integration.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux2]\\label{MellinInversion_aux2}\\lean{MellinInversion_aux2}\\leanok\nLet $f$ be twice differentiable on $(0,\\infty)$, and assume that $f'(x)x^s\\to 0$ as $x\\to 0$, and\nthat $f'(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f'(x)x^{s} dx = -\\int_0^\\infty f''(x)x^{s+1}\\frac{1}{(s+1)}dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux2 {f : \u211d \u2192 \u2102} (s : \u2102) (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (fDiff2 : DifferentiableOn \u211d (deriv f) (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 deriv f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 deriv f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, (deriv f x) * x ^ s =\n      -\u222b x in Ioi 0, (deriv (deriv f) x) * x ^ (s + 1) / (s + 1) := by\n  sorry\n/-%\n\\begin{proof}\n\\uses{PartialIntegration, MellinInversion_aux1}\nPartial integration. (Apply Lemma \\ref{MellinInversion_aux1} to the function $f'$ and $s+1$.)\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux3]%\\label{MellinInversion_aux3}\\lean{MellinInversion_aux3}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen the map  $(x,s) \\mapsto f(x)x^s/(s(s+1))$ is absolutely integrable on\n$(0,\\infty)\\times\\{\\Re s = \\sigma\\}$ for any $\\sigma>0$.\n\\end{lemma}\n%-/\nlemma MellinInversion_aux3 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    IntegrableOn (fun (\u27e8x, t\u27e9 : \u211d \u00d7 \u211d) \u21a6 f x * x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * ((\u03c3 + t * I) + 1)))\n      ((Ioi 0).prod (univ : Set \u211d)) := by\n  sorry\n/-%\n\\begin{proof}\nPut absolute values and estimate.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux4]%\\label{MellinInversion_aux4}\\lean{MellinInversion_aux4}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen we can interchange orders of integration\n$$\n\\int_{(\\sigma)}\\int_0^\\infty f(x)x^{s+1}\\frac{1}{s(s+1)}dx ds =\n\\int_0^\\infty\n\\int_{(\\sigma)}f(x)x^{s+1}\\frac{1}{s(s+1)}ds dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux4 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    VerticalIntegral (fun s \u21a6 \u222b x in Ioi 0, f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 =\n      \u222b x in Ioi 0, VerticalIntegral (fun s \u21a6 f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 := by\n  sorry -- `MeasureTheory.integral_prod` and `MeasureTheory.integral_swap` should be useful here\n/-%\n\\begin{proof}\n\\uses{MellinInversion_aux3}\nFubini-Tonelli.\n\\end{proof}\n%-/\n\nlemma MellinTransform_eq : \ud835\udcdc = mellin := by unfold mellin MellinTransform; simp_rw [smul_eq_mul, mul_comm]\n\nlemma MellinInverseTransform_eq (\u03c3 : \u211d) (f : \u2102 \u2192 \u2102) :\n    MellinInverseTransform f \u03c3 = mellinInv \u03c3 f := by\n  unfold mellinInv MellinInverseTransform VerticalIntegral' VerticalIntegral\n  beta_reduce; ext x\n  rw [\u2190 smul_assoc, smul_eq_mul (a' := I), div_mul]; simp\n\n/-%%\n\\begin{theorem}[MellinInversion]\\label{MellinInversion}\\lean{MellinInversion}\\leanok\nLet $f$ be a twice differentiable function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$, and\nlet $\\sigma$\nbe sufficiently large. Then\n$$f(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}\\mathcal{M}(f)(s)x^{-s}ds.$$\n\\end{theorem}\n\n%[Note: How ``nice''? Schwartz (on $(0,\\infty)$) is certainly enough. As we formalize\n%this, we can add whatever\n% conditions are necessary for the proof to go through.]\n%%-/\ntheorem MellinInversion (\u03c3 : \u211d) {f : \u211d \u2192 \u2102} {x : \u211d} (hx : 0 < x) (hf : MellinConvergent f \u03c3)\n    (hFf : VerticalIntegrable (mellin f) \u03c3) (hfx : ContinuousAt f x) :\n    MellinInverseTransform (\ud835\udcdc f) \u03c3 x = f x := by\n  rw [MellinTransform_eq, MellinInverseTransform_eq, mellin_inversion \u03c3 f hx hf hFf hfx]\n/-%%\n\\begin{proof}\\leanok\n\\uses{PartialIntegration, formulaLtOne, formulaGtOne, MellinTransform,\nMellinInverseTransform, PerronInverseMellin_gt, PerronInverseMellin_lt}\n%MellinInversion_aux1, MellinInversion_aux2, MellinInversion_aux3,\n%MellinInversion_aux4, }\nThe proof is from [Goldfeld-Kontorovich 2012].\nIntegrate by parts twice (assuming $f$ is twice differentiable, and all occurring\nintegrals converge absolutely, and\nboundary terms vanish).\n$$\n\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx = - \\int_0^\\infty f'(x)x^s\\frac{1}{s}dx\n= \\int_0^\\infty f''(x)x^{s+1}\\frac{1}{s(s+1)}dx.\n$$\nWe now have at least quadratic decay in $s$ of the Mellin transform. Inserting this\nformula into the inversion formula and Fubini-Tonelli (we now have absolute\nconvergence!) gives:\n$$\nRHS = \\frac{1}{2\\pi i}\\left(\\int_{(\\sigma)}\\int_0^\\infty\n  f''(t)t^{s+1}\\frac{1}{s(s+1)}dt\\right) x^{-s}ds\n$$\n$$\n= \\int_0^\\infty f''(t) t \\left( \\frac{1}{2\\pi i}\n\\int_{(\\sigma)}(t/x)^s\\frac{1}{s(s+1)}ds\\right) dt.\n$$\nApply the Perron formula to the inside:\n$$\n= \\int_x^\\infty f''(t) t \\left(1-\\frac{x}{t}\\right)dt\n= -\\int_x^\\infty f'(t) dt\n= f(x),\n$$\nwhere we integrated by parts (undoing the first partial integration), and finally\napplied the fundamental theorem of calculus (undoing the second).\n\\end{proof}\n%%-/\n\n\n/-%%\nFinally, we need Mellin Convolutions and properties thereof.\n\\begin{definition}[MellinConvolution]\\label{MellinConvolution}\\lean{MellinConvolution}\n\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. Then we define the\nMellin convolution of $f$ and $g$ to be the function $f\\ast g$ from $\\mathbb{R}_{>0}$\nto $\\mathbb{C}$ defined by\n$$(f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}.$$\n\\end{definition}\n%%-/\nnoncomputable def MellinConvolution (f g : \u211d \u2192 \ud835\udd42) (x : \u211d) : \ud835\udd42 :=\n  \u222b y in Ioi 0, f y * g (x / y) / y\n\n/-%%\nLet us start with a simple property of the Mellin convolution.\n\\begin{lemma}[MellinConvolutionSymmetric]\\label{MellinConvolutionSymmetric}\n\\lean{MellinConvolutionSymmetric}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{R}$ or $\\mathbb{C}$, for $x\\neq0$,\n$$\n  (f\\ast g)(x)=(g\\ast f)(x)\n  .\n$$\n\\end{lemma}\n%%-/\nlemma MellinConvolutionSymmetric (f g : \u211d \u2192 \ud835\udd42) {x : \u211d} (xpos: 0 < x) :\n    MellinConvolution f g x = MellinConvolution g f x := by\n  unfold MellinConvolution\n  calc\n    _ = \u222b y in Ioi 0, f (y * x) * g (1 / y) / y := ?_\n    _ = _ := ?_\n  \u00b7 rw [\u2190 integral_comp_mul_right_I0i_haar (fun y \u21a6 f y * g (x / y)) xpos]\n    simp [div_mul_cancel_right\u2080 <| ne_of_gt xpos]\n  \u00b7 convert (integral_comp_inv_I0i_haar fun y \u21a6 f (y * x) * g (1 / y)).symm using 3\n    rw [one_div_one_div, mul_comm, mul_comm_div, one_mul]\n/-%%\n\\begin{proof}\\leanok\n  \\uses{MellinConvolution}\n  By Definition \\ref{MellinConvolution},\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}\n  $$\n  in which we change variables to $z=x/y$:\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(x/z)g(z)\\frac{dz}{z}\n    =(g\\ast f)(x)\n    .\n  $$\n\\end{proof}\n%%-/\n\n/-%%\nThe Mellin transform of a convolution is the product of the Mellin transforms.\n\\begin{theorem}[MellinConvolutionTransform]\\label{MellinConvolutionTransform}\n\\lean{MellinConvolutionTransform}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ such that\n\\begin{equation}\n  (x,y)\\mapsto f(y)\\frac{g(x/y)}yx^{s-1}\n  \\label{eq:assm_integrable_Mconv}\n\\end{equation}\nis absolutely integrable on $[0,\\infty)^2$.\nThen\n$$\\mathcal{M}(f\\ast g)(s) = \\mathcal{M}(f)(s)\\mathcal{M}(g)(s).$$\n\\end{theorem}\n%%-/\nlemma MellinConvolutionTransform (f g : \u211d \u2192 \u2102) (s : \u2102)\n    (hf : IntegrableOn (fun x y \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)).uncurry\n      (Ioi 0 \u00d7\u02e2 Ioi 0)) :\n    \ud835\udcdc (MellinConvolution f g) s = \ud835\udcdc f s * \ud835\udcdc g s := by\n  dsimp [MellinTransform, MellinConvolution]\n  set f\u2081 : \u211d \u00d7 \u211d \u2192 \u2102 := fun \u27e8x, y\u27e9 \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, \u222b (y : \u211d) in Ioi 0, f\u2081 (x, y) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f\u2081 (x, y) := set_integral_integral_swap _ hf\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x / y) / \u2191y * \u2191x ^ (s - 1) := rfl\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x * y / y) / \u2191y * \u2191(x * y) ^ (s - 1) * y := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * (g x * \u2191x ^ (s - 1)) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * \u222b (x : \u211d) in Ioi 0, g x * \u2191x ^ (s - 1) := ?_\n    _ = _ := integral_mul_right _ _\n  <;> try (rw [set_integral_congr (by simp)]; intro y hy; simp only [ofReal_mul])\n  \u00b7 simp only [integral_mul_right]; rfl\n  \u00b7 simp only [integral_mul_right]\n    have := integral_comp_mul_right_Ioi (fun x \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)) 0 hy\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := slitPlane_ne_zero (Or.inl hy)\n    field_simp at this \u22a2\n    rw [this]\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro x hx\n    have y_ne_zero\u211d : y \u2260 0 := ne_of_gt (mem_Ioi.mp hy)\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := by exact_mod_cast y_ne_zero\u211d\n    field_simp [mul_cpow_ofReal_nonneg (LT.lt.le hx) (LT.lt.le hy)]\n    ring\n  \u00b7 apply integral_mul_left\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform,MellinConvolution}\nBy Definitions \\ref{MellinTransform} and \\ref{MellinConvolution}\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}\\frac{dy}ydx\n$$\nBy (\\ref{eq:assm_integrable_Mconv}) and Fubini's theorem,\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}dx\\frac{dy}y\n$$\nin which we change variables from $x$ to $z=x/y$:\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(z)y^{s-1}z^{s-1}dzdy\n$$\nwhich, by Definition \\ref{MellinTransform}, is\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\mathcal M(f)(s)\\mathcal M(g)(s)\n  .\n$$\n\n\\end{proof}\n%%-/\n\n/-%%\nLet $\\psi$ be a bumpfunction.\n\\begin{theorem}[SmoothExistence]\\label{SmoothExistence}\\lean{SmoothExistence}\\leanok\nThere exists a smooth (once differentiable would be enough), nonnegative ``bumpfunction'' $\\psi$,\n supported in $[1/2,2]$ with total mass one:\n$$\n\\int_0^\\infty \\psi(x)\\frac{dx}{x} = 1.\n$$\n\\end{theorem}\n%%-/\n\nattribute [- simp] one_div in\n\nlemma SmoothExistence : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n    \u03a8.support \u2286 Icc (1 / 2) 2 \u2227 \u222b x in Ici 0, \u03a8 x / x = 1 := by\n  suffices h : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n      \u03a8.support \u2286 Set.Icc (1 / 2) 2 \u2227 0 < \u222b x in Set.Ici 0, \u03a8 x / x by\n    rcases h with \u27e8\u03a8, h\u03a8, h\u03a8nonneg, h\u03a8supp, h\u03a8pos\u27e9\n    let c := (\u222b x in Ici 0, \u03a8 x / x)\n    use fun y \u21a6 \u03a8 y / c\n    refine \u27e8h\u03a8.div_const c, fun y \u21a6 div_nonneg (h\u03a8nonneg y) (le_of_lt h\u03a8pos), ?_, ?_\u27e9\n    \u00b7 rw [Function.support_div, Function.support_const (ne_of_lt h\u03a8pos).symm, inter_univ]\n      convert h\u03a8supp\n    \u00b7 simp only [div_right_comm _ c _, integral_div c, div_self <| ne_of_gt h\u03a8pos]\n\n  have := smooth_urysohn_support_Ioo (a := 1 / 2) (b := 1) (c := 3/2) (d := 2) (by linarith)\n    (by linarith)\n  rcases this with \u27e8\u03a8, h\u03a8ContDiff, _, h\u03a80, h\u03a81, h\u03a8Support\u27e9\n  use \u03a8, h\u03a8ContDiff\n  unfold indicator at h\u03a80 h\u03a81\n  simp only [mem_Icc, Pi.one_apply, Pi.le_def, mem_Ioo] at h\u03a80 h\u03a81\n  simp only [h\u03a8Support, subset_def, mem_Ioo, mem_Icc, and_imp]\n  split_ands\n  \u00b7 exact fun x \u21a6 le_trans (by simp [apply_ite]) (h\u03a80 x)\n  \u00b7 exact fun y hy hy' \u21a6 \u27e8by linarith, by linarith\u27e9\n  \u00b7 rw [integral_pos_iff_support_of_nonneg]\n    \u00b7 simp only [Function.support_div, measurableSet_Ici, Measure.restrict_apply', h\u03a8Support, Function.support_id]\n      have : (Ioo (1 / 2 : \u211d) 2 \u2229 (Iio 0 \u222a Ioi 0) \u2229 Ici 0) = Ioo (1 / 2) 2 := by\n        ext x\n        simp only [mem_inter_iff, mem_Ioo, mem_Ici, mem_Iio, mem_Ioi,\n          mem_union, not_lt, and_true, not_le]\n        constructor\n        \u00b7 exact fun h \u21a6 h.left.left\n        \u00b7 intro h\n          simp only [h, and_self, lt_or_lt_iff_ne, ne_eq, true_and]\n          constructor <;> linarith [h.left]\n      simp only [this, volume_Ioo, ENNReal.ofReal_pos, sub_pos, gt_iff_lt]\n      linarith\n    \u00b7 simp_rw [Pi.le_def, Pi.zero_apply]\n      intro y\n      by_cases h : y \u2208 Function.support \u03a8\n      . apply div_nonneg <| le_trans (by simp [apply_ite]) (h\u03a80 y)\n        rw [h\u03a8Support, mem_Ioo] at h\n        linarith [h.left]\n      . simp only [Function.mem_support, ne_eq, not_not] at h\n        simp [h]\n    \u00b7 have : (fun x \u21a6 \u03a8 x / x).support \u2286 Icc (1 / 2) 2 := by\n        rw [Function.support_div, h\u03a8Support]\n        apply subset_trans (by apply inter_subset_left) Ioo_subset_Icc_self\n      apply (integrableOn_iff_integrable_of_support_subset this).mp\n      apply ContinuousOn.integrableOn_compact isCompact_Icc\n      apply ContinuousOn.div h\u03a8ContDiff.continuous.continuousOn continuousOn_id ?_\n      simp only [mem_Icc, ne_eq, and_imp, id_eq]\n      intros\n      linarith\n/-%%\n\\begin{proof}\\leanok\n\\uses{smooth-ury}\nSame idea as Urysohn-type argument.\n\\end{proof}\n%%-/\n\nlemma mem_within_strip (\u03c3\u2081 \u03c3\u2082 : \u211d) :\n  {s : \u2102 | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} \u2208 \ud835\udcdf {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} := by simp\n\n", "theoremStatement": "lemma MellinOfPsi_aux {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {s : \u2102} (hs : s \u2260 0) :\n    \u222b (x : \u211d) in Ioi 0, (\u03a8 x) * (x : \u2102) ^ (s - 1) =\n    - (1 / s) * \u222b (x : \u211d) in Ioi 0, (deriv \u03a8 x) * (x : \u2102) ^ s ", "theoremName": "MellinOfPsi_aux", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "7f7c6baabc188b7da3afeabf93af687864b56f7a", "date": "2024-03-30"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/MellinCalculus.lean", "module": "PrimeNumberTheoremAnd.MellinCalculus", "jsonFile": "PrimeNumberTheoremAnd.MellinCalculus.jsonl", "positionMetadata": {"lineInFile": 763, "tokenPositionInFile": 32096, "theoremPositionInFile": 56}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 278, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let g {s : \u2102} := fun (x : \u211d)  \u21a6 x ^ s / s\n  have gderiv {s : \u2102} (hs : s \u2260 0) {x: \u211d} (hx : x \u2208 Ioi 0) :\n      deriv g x = x ^ (s - 1) := by\n    have := HasDerivAt.cpow_const (c := s) (hasDerivAt_id (x : \u2102)) (Or.inl hx)\n    simp_rw [mul_one, id_eq] at this\n    rw [deriv_div_const, deriv.comp_ofReal (e := fun x \u21a6 x ^ s)]\n    \u00b7 rw [this.deriv, mul_div_right_comm, div_self hs, one_mul]\n    \u00b7 apply hasDerivAt_deriv_iff.mp\n      simp only [this.deriv, this]\n  calc\n    _ =  \u222b (x : \u211d) in Ioi 0, \u2191(\u03a8 x) * deriv (@g s) x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv (fun x \u21a6 \u2191(\u03a8 x)) x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * x ^ s / s := by simp only [mul_div, g]\n    _ = _ := ?_\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro _ hx\n    simp only [gderiv hs hx]\n  \u00b7 apply PartialIntegration_of_support_in_Icc (\u03a8 \u00b7) g\n      (a := 1 / 2) (b := 2) (by norm_num) (by norm_num)\n    \u00b7 simpa only [Function.support_subset_iff, ne_eq, ofReal_eq_zero]\n    \u00b7 exact (Differentiable.ofReal_comp_iff.mpr (diff\u03a8.differentiable (by norm_num))).differentiableOn\n    \u00b7 refine DifferentiableOn.div_const ?_ s\n      intro a ha\n      refine DifferentiableAt.differentiableWithinAt ?_\n      apply DifferentiableAt.comp_ofReal (e := fun x \u21a6 x ^ s)\n      apply DifferentiableAt.cpow differentiableAt_id' <| differentiableAt_const s\n      exact Or.inl ha\n    \u00b7 simp only [deriv.ofReal_comp']\n      apply Continuous.continuousOn\n      apply Continuous.comp (g := ofReal') continuous_ofReal <| diff\u03a8.continuous_deriv (by norm_num)\n    \u00b7 apply ContinuousOn.congr (f := fun (x : \u211d) \u21a6 (x : \u2102) ^ (s - 1)) ?_ fun x hx \u21a6 gderiv hs hx\n      refine ContinuousOn.cpow ?_ continuousOn_const (by simp)\n      exact Continuous.continuousOn (by continuity)\n  \u00b7 congr; funext; congr\n    apply (hasDerivAt_deriv_iff.mpr ?_).ofReal_comp.deriv\n    exact diff\u03a8.contDiffAt.differentiableAt (by norm_num)\n  \u00b7 simp only [neg_mul, neg_inj]\n    conv => lhs; rhs; intro; rw [\u2190 mul_one_div, mul_comm]\n    rw [integral_mul_left]", "proofType": "tactic", "proofLengthLines": 40, "proofLengthTokens": 2043}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\n", "theoremStatement": "lemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) ", "theoremName": "Asymptotics.IsBigO.sq", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "8ac305dd56b39a62a48a98ec84c04f4486a7ee08", "date": 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"Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", 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"PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simpa [pow_two] using h.mul h", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 37}}
+{"srcContext": "/-\nCopyright (c) 2024 Lawrence Wu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Lawrence Wu\n-/\n\nimport Mathlib.MeasureTheory.Integral.SetIntegral\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta\n\n/-!\n# Uniform Asymptotics\n\nFor a family of functions `f : \u03b9 \u00d7 \u03b1 \u2192 E` and `g : \u03b1 \u2192 E`, we can think of\n`f =O[\ud835\udcdf s \u00d7\u02e2 l] fun (i, x) \u21a6 g x` as expressing that `f i` is O(g) uniformly on `s`.\n\nThis file provides methods for constructing `=O[\ud835\udcdf s \u00d7\u02e2 l]` relations (similarly `\u0398`)\nand deriving their consequences.\n-/\n\nopen Filter\n\nopen Topology\n\nnamespace Asymptotics\n\nvariable {\u03b1 \u03b9 E F : Type*} {s : Set \u03b9}\n\nsection Basic\n\nvariable [Norm E] [Norm F] {f : \u03b9 \u00d7 \u03b1 \u2192 E} {g : \u03b1 \u2192 F} {l : Filter \u03b1}\n\n/-- If f = O(g) uniformly on `s`, then f_i = O(g) for any i.` -/\ntheorem isBigO_of_isBigOUniformly (h : f =O[\ud835\udcdf s \u00d7\u02e2 l] (g \u2218 Prod.snd)) {i : \u03b9} (hi : i \u2208 s) :\n    (fun x \u21a6 f (i, x)) =O[l] g := by\n  obtain \u27e8C, hC\u27e9 := h.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  refine isBigO_iff.mpr \u27e8C, Filter.eventually_iff_exists_mem.mpr \u27e8v, hv, ?_\u27e9\u27e9\n  exact fun y hy \u21a6 ht _ <| huv \u27e8hu hi, hy\u27e9\n\n/-- If f = \u03a9(g) uniformly on `s`, then f_i = \u03a9(g) for any i.` -/\ntheorem isBigO_rev_of_isBigOUniformly_rev (h : (g \u2218 Prod.snd) =O[\ud835\udcdf s \u00d7\u02e2 l] f) {i : \u03b9} (hi : i \u2208 s) :\n    g =O[l] fun x \u21a6 f (i, x) := by\n  obtain \u27e8C, hC\u27e9 := h.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  refine isBigO_iff.mpr \u27e8C, Filter.eventually_iff_exists_mem.mpr \u27e8v, hv, ?_\u27e9\u27e9\n  exact fun y hy \u21a6 ht (i, y) <| huv \u27e8hu hi, hy\u27e9\n\n/-- If f = \u0398(g) uniformly on `s`, then f_i = \u0398(g) for any i.` -/\ntheorem isTheta_of_isThetaUniformly (h : f =\u0398[\ud835\udcdf s \u00d7\u02e2 l] (g \u2218 Prod.snd)) {i : \u03b9} (hi : i \u2208 s) :\n    (fun x \u21a6 f (i, x)) =\u0398[l] g :=\n  \u27e8isBigO_of_isBigOUniformly h.1 hi, isBigO_rev_of_isBigOUniformly_rev h.2 hi\u27e9\n\nend Basic\n\nsection Order\n\nvariable [NormedAddCommGroup \u03b1] [LinearOrder \u03b1] [ProperSpace \u03b1] [NormedAddCommGroup F]\n\ntheorem isLittleO_const_fst_atBot [NoMinOrder \u03b1] [ClosedIicTopology \u03b1] (c : F) (ly : Filter E) :\n    (fun (_ : \u03b1 \u00d7 E) \u21a6 c) =o[atBot \u00d7\u02e2 ly] Prod.fst := by\n  refine ly.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.fst) =o[atBot \u00d7\u02e2 ly] (id \u2218 Prod.fst)\n  rewrite [\u2190 isLittleO_map, map_fst_prod]\n  exact isLittleO_const_id_atBot2 c\n\ntheorem isLittleO_const_snd_atBot [NoMinOrder \u03b1] [ClosedIicTopology \u03b1] (c : F) (lx : Filter E) :\n    (fun (_ : E \u00d7 \u03b1) \u21a6 c) =o[lx \u00d7\u02e2 atBot] Prod.snd := by\n  refine lx.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.snd) =o[lx \u00d7\u02e2 atBot] (id \u2218 Prod.snd)\n  rewrite [\u2190 isLittleO_map, map_snd_prod]\n  exact isLittleO_const_id_atBot2 c\n\ntheorem isLittleO_const_fst_atTop [NoMaxOrder \u03b1] [ClosedIciTopology \u03b1] (c : F) (ly : Filter E) :\n    (fun (_ : \u03b1 \u00d7 E) \u21a6 c) =o[atTop \u00d7\u02e2 ly] Prod.fst := by\n  refine ly.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.fst) =o[atTop \u00d7\u02e2 ly] (id \u2218 Prod.fst)\n  rewrite [\u2190 isLittleO_map, map_fst_prod]\n  exact isLittleO_const_id_atTop2 c\n\ntheorem isLittleO_const_snd_atTop [NoMaxOrder \u03b1] [ClosedIciTopology \u03b1] (c : F) (lx : Filter E) :\n    (fun (_ : E \u00d7 \u03b1) \u21a6 c) =o[lx \u00d7\u02e2 atTop] Prod.snd := by\n  refine lx.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.snd) =o[lx \u00d7\u02e2 atTop] (id \u2218 Prod.snd)\n  rewrite [\u2190 isLittleO_map, map_snd_prod]\n  exact isLittleO_const_id_atTop2 c\n\nend Order\n\nsection ContinuousOn\n\nvariable [TopologicalSpace \u03b9] {C : \u03b9 \u2192 E} {c : F}\n\nsection IsBigO\n\nvariable [SeminormedAddGroup E] [Norm F]\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly bounded w.r.t. `s` by\n`\u2a06 i \u2208 s, \u2016C i\u2016`, if `s` is compact and `C` is continuous. -/\ntheorem _root_.ContinuousOn.const_isBigOWithUniformlyOn_isCompact\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hc : \u2016c\u2016 \u2260 0) (l : Filter \u03b1) :\n    IsBigOWith (sSup (norm '' (C '' s)) / \u2016c\u2016) (\ud835\udcdf s \u00d7\u02e2 l)\n    (fun (i, _x) \u21a6 C i) fun _ => c := by\n  refine isBigOWith_iff.mpr <| eventually_of_mem ?_ (fun x hx \u21a6 ?_) (U := s \u00d7\u02e2 Set.univ)\n  \u00b7 exact prod_mem_prod (mem_principal_self s) univ_mem\n  \u00b7 rw [div_mul_cancel\u2080 _ hc]\n    replace hs := hs.image_of_continuousOn hf |>.image continuous_norm\n    have h_sSup := hs.isLUB_sSup <| Set.image_nonempty.mpr <| Set.image_nonempty.mpr \u27e8x.1, hx.1\u27e9\n    exact h_sSup.1 <| Set.mem_image_of_mem _ <| Set.mem_image_of_mem _ hx.1\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly O(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous. -/\ntheorem _root_.ContinuousOn.const_isBigOUniformlyOn_isCompact\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hc : \u2016c\u2016 \u2260 0) (l : Filter \u03b1) :\n    (fun (i, _x) \u21a6 C i) =O[\ud835\udcdf s \u00d7\u02e2 l] fun _ => c :=\n  (hf.const_isBigOWithUniformlyOn_isCompact hs hc l).isBigO\n\nend IsBigO\n\nsection IsTheta\n\nvariable [NormedAddGroup E] [SeminormedAddGroup F]\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly bounded below w.r.t. `s` by\n`\u2293 i \u2208 s, \u2016C i\u2016`, if `s` is compact and `C` is continuous. -/\ntheorem _root_.ContinuousOn.const_isBigOWithUniformlyOn_isCompact_rev\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hC : \u2200 i \u2208 s, C i \u2260 0) (l : Filter \u03b1) :\n    IsBigOWith (\u2016c\u2016 / sInf (norm '' (C '' s))) (\ud835\udcdf s \u00d7\u02e2 l)\n    (fun _ => c) fun (i, _x) \u21a6 C i := by\n  refine isBigOWith_iff.mpr <| eventually_of_mem ?_ (fun x hx \u21a6 ?_) (U := s \u00d7\u02e2 Set.univ)\n  \u00b7 exact prod_mem_prod (mem_principal_self s) univ_mem\n  \u00b7 rewrite [mul_comm_div]\n    replace hs := hs.image_of_continuousOn hf |>.image continuous_norm\n    have h_sInf := hs.isGLB_sInf <| Set.image_nonempty.mpr <| Set.image_nonempty.mpr \u27e8x.1, hx.1\u27e9\n    refine le_mul_of_one_le_right (norm_nonneg c) <| (one_le_div ?_).mpr <|\n      h_sInf.1 <| Set.mem_image_of_mem _ <| Set.mem_image_of_mem _ hx.1\n    obtain \u27e8_, \u27e8x, hx, hCx\u27e9, hnormCx\u27e9 := hs.sInf_mem h_sInf.nonempty\n    rewrite [\u2190 hnormCx, \u2190 hCx]\n    exact (norm_ne_zero_iff.mpr (hC x hx)).symm.lt_of_le (norm_nonneg _)\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly \u03a9(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous with no zeros on `s`. -/\ntheorem _root_.ContinuousOn.const_isBigOUniformlyOn_isCompact_rev\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hC : \u2200 i \u2208 s, C i \u2260 0) (l : Filter \u03b1) :\n    (fun _ => c) =O[\ud835\udcdf s \u00d7\u02e2 l] (fun (i, _x) \u21a6 C i) :=\n  (hf.const_isBigOWithUniformlyOn_isCompact_rev hs hC l).isBigO\n\n", "theoremStatement": "/-- A family of constant functions `f (i, x) = C i` is uniformly \u0398(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous with no zeros on `s`. -/\ntheorem _root_.ContinuousOn.const_isThetaUniformlyOn_isCompact (hf : ContinuousOn C s)\n    (hs : IsCompact s) (hc : \u2016c\u2016 \u2260 0) (hC : \u2200 i \u2208 s, C i \u2260 0) 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"Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  \u27e8hf.const_isBigOUniformlyOn_isCompact hs hc l, hf.const_isBigOUniformlyOn_isCompact_rev hs hC l\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 101}}
+{"srcContext": "import Mathlib.Analysis.Calculus.Deriv.Support\nimport Mathlib.Analysis.Distribution.SchwartzSpace\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\n\nopen Real Complex MeasureTheory Filter Topology BoundedContinuousFunction SchwartzMap  BigOperators\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] {n : \u2115}\n\n@[ext] structure CS (n : \u2115) (E : Type*) [NormedAddCommGroup E] [NormedSpace \u211d E] where\n  toFun : \u211d \u2192 E\n  h1 : ContDiff \u211d n toFun\n  h2 : HasCompactSupport toFun\n\nstructure trunc extends (CS 2 \u211d) where\n  h3 : (Set.Icc (-1) (1)).indicator 1 \u2264 toFun\n  h4 : toFun \u2264 Set.indicator (Set.Ioo (-2) (2)) 1\n\nstructure W1 (n : \u2115) (E : Type*) [NormedAddCommGroup E] [NormedSpace \u211d E] where\n  toFun : \u211d \u2192 E\n  smooth : ContDiff \u211d n toFun\n  integrable : \u2200 \u2983k\u2984, k \u2264 n \u2192 Integrable (iteratedDeriv k toFun)\n\nabbrev W21 := W1 2 \u2102\n\nsection lemmas\n\nnoncomputable def funscale {E : Type*} (g : \u211d \u2192 E) (R x : \u211d) : E := g (R\u207b\u00b9 \u2022 x)\n\nlemma contDiff_ofReal : ContDiff \u211d \u22a4 ofReal' := by\n  have key x : HasDerivAt ofReal' 1 x := hasDerivAt_id x |>.ofReal_comp\n  have key' : deriv ofReal' = fun _ => 1 := by ext x ; exact (key x).deriv\n  refine contDiff_top_iff_deriv.mpr \u27e8fun x => (key x).differentiableAt, ?_\u27e9\n  simpa [key'] using contDiff_const\n\nlemma tendsto_funscale {f : \u211d \u2192 E} (hf : ContinuousAt f 0) (x : \u211d) :\n    Tendsto (fun R => funscale f R x) atTop (\ud835\udcdd (f 0)) :=\n  hf.tendsto.comp (by simpa using tendsto_inv_atTop_zero.mul_const x)\n\nend lemmas\n\nnamespace CS\n\nvariable {f : CS n E} {R x v : \u211d}\n\ninstance : CoeFun (CS n E) (fun _ => \u211d \u2192 E) where coe := CS.toFun\n\ninstance : Coe (CS n \u211d) (CS n \u2102) where coe f := \u27e8fun x => f x,\n  contDiff_ofReal.of_le le_top |>.comp f.h1, f.h2.comp_left (g := ofReal') rfl\u27e9\n\ndef neg (f : CS n E) : CS n E where\n  toFun := -f\n  h1 := f.h1.neg\n  h2 := by simpa [HasCompactSupport, tsupport] using f.h2\n\ninstance : Neg (CS n E) where neg := neg\n\n@[simp] lemma neg_apply {x : \u211d} : (-f) x = - (f x) := rfl\n\ndef smul (R : \u211d) (f : CS n E) : CS n E := \u27e8R \u2022 f, f.h1.const_smul R, f.h2.smul_left\u27e9\n\ninstance : HSMul \u211d (CS n E) (CS n E) where hSMul := smul\n\n@[simp] lemma smul_apply : (R \u2022 f) x = R \u2022 f x := rfl\n\nlemma continuous (f : CS n E) : Continuous f := f.h1.continuous\n\nnoncomputable def deriv (f : CS (n + 1) E) : CS n E where\n  toFun := _root_.deriv f\n  h1 := (contDiff_succ_iff_deriv.mp f.h1).2\n  h2 := f.h2.deriv\n\nlemma hasDerivAt (f : CS (n + 1) E) (x : \u211d) : HasDerivAt f (f.deriv x) x :=\n  (f.h1.differentiable (by simp)).differentiableAt.hasDerivAt\n\nlemma deriv_apply {f : CS (n + 1) E} {x : \u211d} : f.deriv x = _root_.deriv f x := rfl\n\nlemma deriv_smul {f : CS (n + 1) E} : (R \u2022 f).deriv = R \u2022 f.deriv := by\n  ext x ; exact (f.hasDerivAt x |>.const_smul R).deriv\n\nnoncomputable def scale (g : CS n E) (R : \u211d) : CS n E := by\n  by_cases h : R = 0\n  \u00b7 exact \u27e80, contDiff_const, by simp [HasCompactSupport, tsupport]\u27e9\n  \u00b7 refine \u27e8fun x => funscale g R x, ?_, ?_\u27e9\n    \u00b7 exact g.h1.comp (contDiff_const.smul contDiff_id)\n    \u00b7 exact g.h2.comp_smul (inv_ne_zero h)\n\nlemma deriv_scale {f : CS (n + 1) E} : (f.scale R).deriv = R\u207b\u00b9 \u2022 f.deriv.scale R := by\n  ext v ; by_cases hR : R = 0 <;> simp [hR, scale]\n  \u00b7 simp [deriv, smul] ; exact deriv_const _ _\n  \u00b7 exact ((f.hasDerivAt (R\u207b\u00b9 \u2022 v)).scomp v (by simpa using (hasDerivAt_id v).const_smul R\u207b\u00b9)).deriv\n\nlemma deriv_scale' {f : CS (n + 1) E} : (f.scale R).deriv v = R\u207b\u00b9 \u2022 f.deriv (R\u207b\u00b9 \u2022 v) := by\n  rw [deriv_scale, smul_apply]\n  by_cases hR : R = 0 <;> simp [hR, scale, funscale]\n\nlemma hasDerivAt_scale (f : CS (n + 1) E) (R x : \u211d) :\n    HasDerivAt (f.scale R) (R\u207b\u00b9 \u2022 _root_.deriv f (R\u207b\u00b9 \u2022 x)) x := by\n  convert hasDerivAt (f.scale R) x ; rw [deriv_scale'] ; rfl\n\nlemma tendsto_scale (f : CS n E) (x : \u211d) : Tendsto (fun R => f.scale R x) atTop (\ud835\udcdd (f 0)) := by\n  apply (tendsto_funscale f.continuous.continuousAt x).congr'\n  filter_upwards [eventually_ne_atTop 0] with R hR ; simp [scale, hR]\n\nlemma bounded : \u2203 C, \u2200 v, \u2016f v\u2016 \u2264 C := by\n  obtain \u27e8x, hx\u27e9 := (continuous_norm.comp f.continuous).exists_forall_ge_of_hasCompactSupport f.h2.norm\n  exact \u27e8_, hx\u27e9\n\nend CS\n\nnamespace trunc\n\ninstance : CoeFun trunc (fun _ => \u211d \u2192 \u211d) where coe f := f.toFun\n\ninstance : Coe trunc (CS 2 \u211d) where coe := trunc.toCS\n\nlemma nonneg (g : trunc) (x : \u211d) : 0 \u2264 g x := (Set.indicator_nonneg (by simp) x).trans (g.h3 x)\n\nlemma le_one (g : trunc) (x : \u211d) : g x \u2264 1 := (g.h4 x).trans <| Set.indicator_le_self' (by simp) x\n\nlemma zero (g : trunc) : g =\u1da0[\ud835\udcdd 0] 1 := by\n  have : Set.Icc (-1) 1 \u2208 \ud835\udcdd (0 : \u211d) := by apply Icc_mem_nhds <;> linarith\n  exact eventually_of_mem this (fun x hx => le_antisymm (g.le_one x) (by simpa [hx] using g.h3 x))\n\n@[simp] lemma zero_at {g : trunc} : g 0 = 1 := g.zero.eq_of_nhds\n\nend trunc\n\nnamespace W1\n\ninstance : CoeFun (W1 n E) (fun _ => \u211d \u2192 E) where coe := W1.toFun\n\nlemma continuous (f : W1 n E) : Continuous f := f.smooth.continuous\n\nlemma differentiable (f : W1 (n + 1) E) : Differentiable \u211d f :=\n  f.smooth.differentiable (by simp)\n\nlemma iteratedDeriv_sub {f g : \u211d \u2192 E} (hf : ContDiff \u211d n f) (hg : ContDiff \u211d n g) :\n    iteratedDeriv n (f - g) = iteratedDeriv n f - iteratedDeriv n g := by\n  induction n generalizing f g with\n  | zero => rfl\n  | succ n ih =>\n    have hf' : ContDiff \u211d n (deriv f) := hf.iterate_deriv' n 1\n    have hg' : ContDiff \u211d n (deriv g) := hg.iterate_deriv' n 1\n    have hfg : deriv (f - g) = deriv f - deriv g := by\n      ext x ; apply deriv_sub\n      \u00b7 exact (hf.differentiable (by simp)).differentiableAt\n      \u00b7 exact (hg.differentiable (by simp)).differentiableAt\n    simp_rw [iteratedDeriv_succ', \u2190 ih hf' hg', hfg]\n\nnoncomputable def deriv (f : W1 (n + 1) E) : W1 n E where\n  toFun := _root_.deriv f\n  smooth := contDiff_succ_iff_deriv.mp f.smooth |>.2\n  integrable k hk := by\n    simpa [iteratedDeriv_succ'] using f.integrable (Nat.succ_le_succ hk)\n\nlemma hasDerivAt (f : W1 (n + 1) E) (x : \u211d) : HasDerivAt f (f.deriv x) x :=\n  f.differentiable.differentiableAt.hasDerivAt\n\ndef sub (f g : W1 n E) : W1 n E where\n  toFun := f - g\n  smooth := f.smooth.sub g.smooth\n  integrable k hk := by\n    have hf : ContDiff \u211d k f := f.smooth.of_le (by simp [hk])\n    have hg : ContDiff \u211d k g := g.smooth.of_le (by simp [hk])\n    simpa [iteratedDeriv_sub hf hg] using (f.integrable hk).sub (g.integrable hk)\n\ninstance : Sub (W1 n E) where sub := sub\n\n", "theoremStatement": "lemma integrable_iteratedDeriv_Schwarz {f : \ud835\udce2(\u211d, \u2102)} : Integrable (iteratedDeriv n f) ", "theoremName": "W1.integrable_iteratedDeriv_Schwarz", "fileCreated": {"commit": "3556325413630883e7e5edecd71bae0bfd432ab0", "date": "2024-04-17"}, "theoremCreated": {"commit": "3556325413630883e7e5edecd71bae0bfd432ab0", "date": "2024-04-17"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Sobolev.lean", "module": "PrimeNumberTheoremAnd.Sobolev", "jsonFile": "PrimeNumberTheoremAnd.Sobolev.jsonl", "positionMetadata": {"lineInFile": 169, "tokenPositionInFile": 6257, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", 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"Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", 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+{"srcContext": "/-\nCopyright (c) 2024 Lawrence Wu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Lawrence Wu\n-/\n\nimport Mathlib.MeasureTheory.Integral.SetIntegral\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta\n\n/-!\n# Uniform Asymptotics\n\nFor a family of functions `f : \u03b9 \u00d7 \u03b1 \u2192 E` and `g : \u03b1 \u2192 E`, we can think of\n`f =O[\ud835\udcdf s \u00d7\u02e2 l] fun (i, x) \u21a6 g x` as expressing that `f i` is O(g) uniformly on `s`.\n\nThis file provides methods for constructing `=O[\ud835\udcdf s \u00d7\u02e2 l]` relations (similarly `\u0398`)\nand deriving their consequences.\n-/\n\nopen Filter\n\nopen Topology\n\nnamespace Asymptotics\n\nvariable {\u03b1 \u03b9 E F : Type*} {s : Set \u03b9}\n\nsection Basic\n\nvariable [Norm E] [Norm F] {f : \u03b9 \u00d7 \u03b1 \u2192 E} {g : \u03b1 \u2192 F} {l : Filter \u03b1}\n\n/-- If f = O(g) uniformly on `s`, then f_i = O(g) for any i.` -/\ntheorem isBigO_of_isBigOUniformly (h : f =O[\ud835\udcdf s \u00d7\u02e2 l] (g \u2218 Prod.snd)) {i : \u03b9} (hi : i \u2208 s) :\n    (fun x \u21a6 f (i, x)) =O[l] g := by\n  obtain \u27e8C, hC\u27e9 := h.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  refine isBigO_iff.mpr \u27e8C, Filter.eventually_iff_exists_mem.mpr \u27e8v, hv, ?_\u27e9\u27e9\n  exact fun y hy \u21a6 ht _ <| huv \u27e8hu hi, hy\u27e9\n\n/-- If f = \u03a9(g) uniformly on `s`, then f_i = \u03a9(g) for any i.` -/\ntheorem isBigO_rev_of_isBigOUniformly_rev (h : (g \u2218 Prod.snd) =O[\ud835\udcdf s \u00d7\u02e2 l] f) {i : \u03b9} (hi : i \u2208 s) :\n    g =O[l] fun x \u21a6 f (i, x) := by\n  obtain \u27e8C, hC\u27e9 := h.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  refine isBigO_iff.mpr \u27e8C, Filter.eventually_iff_exists_mem.mpr \u27e8v, hv, ?_\u27e9\u27e9\n  exact fun y hy \u21a6 ht (i, y) <| huv \u27e8hu hi, hy\u27e9\n\n/-- If f = \u0398(g) uniformly on `s`, then f_i = \u0398(g) for any i.` -/\ntheorem isTheta_of_isThetaUniformly (h : f =\u0398[\ud835\udcdf s \u00d7\u02e2 l] (g \u2218 Prod.snd)) {i : \u03b9} (hi : i \u2208 s) :\n    (fun x \u21a6 f (i, x)) =\u0398[l] g :=\n  \u27e8isBigO_of_isBigOUniformly h.1 hi, isBigO_rev_of_isBigOUniformly_rev h.2 hi\u27e9\n\nend Basic\n\nsection Order\n\nvariable [NormedAddCommGroup \u03b1] [LinearOrder \u03b1] [ProperSpace \u03b1] [NormedAddCommGroup F]\n\ntheorem isLittleO_const_fst_atBot [NoMinOrder \u03b1] [ClosedIicTopology \u03b1] (c : F) (ly : Filter E) :\n    (fun (_ : \u03b1 \u00d7 E) \u21a6 c) =o[atBot \u00d7\u02e2 ly] Prod.fst := by\n  refine ly.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.fst) =o[atBot \u00d7\u02e2 ly] (id \u2218 Prod.fst)\n  rewrite [\u2190 isLittleO_map, map_fst_prod]\n  exact isLittleO_const_id_atBot2 c\n\ntheorem isLittleO_const_snd_atBot [NoMinOrder \u03b1] [ClosedIicTopology \u03b1] (c : F) (lx : Filter E) :\n    (fun (_ : E \u00d7 \u03b1) \u21a6 c) =o[lx \u00d7\u02e2 atBot] Prod.snd := by\n  refine lx.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.snd) =o[lx \u00d7\u02e2 atBot] (id \u2218 Prod.snd)\n  rewrite [\u2190 isLittleO_map, map_snd_prod]\n  exact isLittleO_const_id_atBot2 c\n\ntheorem isLittleO_const_fst_atTop [NoMaxOrder \u03b1] [ClosedIciTopology \u03b1] (c : F) (ly : Filter E) :\n    (fun (_ : \u03b1 \u00d7 E) \u21a6 c) =o[atTop \u00d7\u02e2 ly] Prod.fst := by\n  refine ly.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.fst) =o[atTop \u00d7\u02e2 ly] (id \u2218 Prod.fst)\n  rewrite [\u2190 isLittleO_map, map_fst_prod]\n  exact isLittleO_const_id_atTop2 c\n\ntheorem isLittleO_const_snd_atTop [NoMaxOrder \u03b1] [ClosedIciTopology \u03b1] (c : F) (lx : Filter E) :\n    (fun (_ : E \u00d7 \u03b1) \u21a6 c) =o[lx \u00d7\u02e2 atTop] Prod.snd := by\n  refine lx.eq_or_neBot.casesOn (fun h \u21a6 by simp [h]) (fun _ \u21a6 ?_)\n  show ((fun _ \u21a6 c) \u2218 Prod.snd) =o[lx \u00d7\u02e2 atTop] (id \u2218 Prod.snd)\n  rewrite [\u2190 isLittleO_map, map_snd_prod]\n  exact isLittleO_const_id_atTop2 c\n\nend Order\n\nsection ContinuousOn\n\nvariable [TopologicalSpace \u03b9] {C : \u03b9 \u2192 E} {c : F}\n\nsection IsBigO\n\nvariable [SeminormedAddGroup E] [Norm F]\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly bounded w.r.t. `s` by\n`\u2a06 i \u2208 s, \u2016C i\u2016`, if `s` is compact and `C` is continuous. -/\ntheorem _root_.ContinuousOn.const_isBigOWithUniformlyOn_isCompact\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hc : \u2016c\u2016 \u2260 0) (l : Filter \u03b1) :\n    IsBigOWith (sSup (norm '' (C '' s)) / \u2016c\u2016) (\ud835\udcdf s \u00d7\u02e2 l)\n    (fun (i, _x) \u21a6 C i) fun _ => c := by\n  refine isBigOWith_iff.mpr <| eventually_of_mem ?_ (fun x hx \u21a6 ?_) (U := s \u00d7\u02e2 Set.univ)\n  \u00b7 exact prod_mem_prod (mem_principal_self s) univ_mem\n  \u00b7 rw [div_mul_cancel\u2080 _ hc]\n    replace hs := hs.image_of_continuousOn hf |>.image continuous_norm\n    have h_sSup := hs.isLUB_sSup <| Set.image_nonempty.mpr <| Set.image_nonempty.mpr \u27e8x.1, hx.1\u27e9\n    exact h_sSup.1 <| Set.mem_image_of_mem _ <| Set.mem_image_of_mem _ hx.1\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly O(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous. -/\ntheorem _root_.ContinuousOn.const_isBigOUniformlyOn_isCompact\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hc : \u2016c\u2016 \u2260 0) (l : Filter \u03b1) :\n    (fun (i, _x) \u21a6 C i) =O[\ud835\udcdf s \u00d7\u02e2 l] fun _ => c :=\n  (hf.const_isBigOWithUniformlyOn_isCompact hs hc l).isBigO\n\nend IsBigO\n\nsection IsTheta\n\nvariable [NormedAddGroup E] [SeminormedAddGroup F]\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly bounded below w.r.t. `s` by\n`\u2293 i \u2208 s, \u2016C i\u2016`, if `s` is compact and `C` is continuous. -/\ntheorem _root_.ContinuousOn.const_isBigOWithUniformlyOn_isCompact_rev\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hC : \u2200 i \u2208 s, C i \u2260 0) (l : Filter \u03b1) :\n    IsBigOWith (\u2016c\u2016 / sInf (norm '' (C '' s))) (\ud835\udcdf s \u00d7\u02e2 l)\n    (fun _ => c) fun (i, _x) \u21a6 C i := by\n  refine isBigOWith_iff.mpr <| eventually_of_mem ?_ (fun x hx \u21a6 ?_) (U := s \u00d7\u02e2 Set.univ)\n  \u00b7 exact prod_mem_prod (mem_principal_self s) univ_mem\n  \u00b7 rewrite [mul_comm_div]\n    replace hs := hs.image_of_continuousOn hf |>.image continuous_norm\n    have h_sInf := hs.isGLB_sInf <| Set.image_nonempty.mpr <| Set.image_nonempty.mpr \u27e8x.1, hx.1\u27e9\n    refine le_mul_of_one_le_right (norm_nonneg c) <| (one_le_div ?_).mpr <|\n      h_sInf.1 <| Set.mem_image_of_mem _ <| Set.mem_image_of_mem _ hx.1\n    obtain \u27e8_, \u27e8x, hx, hCx\u27e9, hnormCx\u27e9 := hs.sInf_mem h_sInf.nonempty\n    rewrite [\u2190 hnormCx, \u2190 hCx]\n    exact (norm_ne_zero_iff.mpr (hC x hx)).symm.lt_of_le (norm_nonneg _)\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly \u03a9(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous with no zeros on `s`. -/\ntheorem _root_.ContinuousOn.const_isBigOUniformlyOn_isCompact_rev\n    (hf : ContinuousOn C s) (hs : IsCompact s) (hC : \u2200 i \u2208 s, C i \u2260 0) (l : Filter \u03b1) :\n    (fun _ => c) =O[\ud835\udcdf s \u00d7\u02e2 l] (fun (i, _x) \u21a6 C i) :=\n  (hf.const_isBigOWithUniformlyOn_isCompact_rev hs hC l).isBigO\n\n/-- A family of constant functions `f (i, x) = C i` is uniformly \u0398(1) w.r.t. `s`,\nif `s` is compact and `C` is continuous with no zeros on `s`. -/\ntheorem _root_.ContinuousOn.const_isThetaUniformlyOn_isCompact (hf : ContinuousOn C s)\n    (hs : IsCompact s) (hc : \u2016c\u2016 \u2260 0) (hC : \u2200 i \u2208 s, C i \u2260 0) (l : Filter \u03b1) :\n    (fun (i, _x) \u21a6 C i) =\u0398[\ud835\udcdf s \u00d7\u02e2 l] fun _ => c :=\n  \u27e8hf.const_isBigOUniformlyOn_isCompact hs hc l, hf.const_isBigOUniformlyOn_isCompact_rev hs hC l\u27e9\n\nend IsTheta\n\nend ContinuousOn\n\nsection Integral -- TODO: move to Integral/Asymptotics.lean\n\nopen MeasureTheory\n\nvariable [MeasurableSpace \u03b9] [NormedAddCommGroup E] {f : \u03b9 \u00d7 \u03b1 \u2192 E} {g : \u03b1 \u2192 F}\n  {\u03bc : Measure \u03b9} {l : Filter \u03b1}\n\n", "theoremStatement": "/-- Let `f : X x Y \u2192 Z`. 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"Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8C, hC\u27e9 := hf.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  obtain \u27e8w, hwl, hw\u27e9 := hfm.exists_mem\n  refine eventually_iff_exists_mem.mpr \u27e8w \u2229 v, inter_mem hwl hv, fun x hx \u21a6 ?_\u27e9\n  haveI : IsFiniteMeasure (\u03bc.restrict s) :=\n    \u27e8by convert h\u03bc using 1; exact Measure.restrict_apply_univ s\u27e9\n  refine Integrable.mono' (integrable_const (C * \u2016g x\u2016)) (hw x hx.1) ?_\n  filter_upwards [MeasureTheory.self_mem_ae_restrict hs]\n  intro y hy\n  exact ht (y, x) <| huv \u27e8hu hy, hx.2\u27e9", "proofType": "tactic", "proofLengthLines": 11, "proofLengthTokens": 542}}
+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\ntheorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) := by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz\n\ntheorem primesBetween_le_siftedSum_add :\n    primesBetween x (x+y) \u2264 (primeInterSieve x y z hz).siftedSum + z := by\n  classical\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a (Finset.Icc 1 (Nat.floor z))).card\n  \u00b7 rw[primesBetween]\n    norm_cast\n    apply Finset.card_le_card\n    apply primesBetween_subset _ _ _ hx\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card\n    + \u2191(Finset.Icc 1 (Nat.floor z)).card\n  \u00b7 norm_cast\n    apply Finset.card_union_le\n  rw[siftedSum_eq_card]\n  gcongr\n  rw[Nat.card_Icc]\n  simp\n  apply Nat.floor_le\n  linarith\n\nsection Remainder\n\ntheorem Ioc_filter_dvd_eq (d a b: \u2115) (hd : d \u2260 0) :\n  Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b) =\n    Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d)) := by\n  ext n\n  simp only [Finset.mem_filter, Finset.mem_Ioc, Nat.ceil_le, Finset.mem_image,\n    Nat.le_floor_iff (show 0 \u2264 x+y by linarith)]\n  constructor\n  \u00b7 intro hn\n    use  n/d\n    rcases hn with \u27e8\u27e8han, hnb\u27e9, hd\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 exact Nat.div_lt_div_of_lt_of_dvd hd han\n    \u00b7 exact Nat.div_le_div_right (Nat.le_floor hnb)\n    \u00b7 exact Nat.div_mul_cancel hd\n  \u00b7 rintro \u27e8r, \u27e8ha, ha'\u27e9, rfl\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 refine (Nat.div_lt_iff_lt_mul ?_).mp ha\n      omega\n    \u00b7 exact Nat.mul_le_of_le_div d r b ha'\n    \u00b7 exact Nat.dvd_mul_left d r\n\ntheorem card_Ioc_filter_dvd (d a b: \u2115) (hd : d \u2260 0) :\n    (Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b)).card = b / d - a / d  := by\n  rw [Ioc_filter_dvd_eq _ _ _ hd]\n  rw [Finset.card_image_of_injective _ <| mul_left_injective\u2080 hd]\n  simp\n\ntheorem multSum_eq (d : \u2115) (hd : d \u2260 0):\n    (primeInterSieve x y z hz).multSum d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) := by\n  unfold Sieve.multSum\n  rw[primeInterSieve]\n  simp\n  trans \u2191(Finset.Ioc (Nat.ceil x - 1) (Nat.floor (x+y)) |>.filter (d \u2223 \u00b7) |>.card)\n  \u00b7 rw [\u2190Nat.Icc_succ_left]\n    congr\n    rw [\u2190Nat.succ_sub]; rfl\n    simp [hx]\n  \u00b7 rw[BrunTitchmarsh.card_Ioc_filter_dvd _ _ _ hd]\n\ntheorem rem_eq (d : \u2115) (hd : d \u2260 0) : (primeInterSieve x y z hz).rem d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) - (\u2191d)\u207b\u00b9 * y := by\n  unfold Sieve.rem\n  rw[multSum_eq x y z hx hz d hd]\n  simp [primeInterSieve, if_neg hd]\n\n", "theoremStatement": "theorem Nat.ceil_le_self_add_one (x : \u211d) (hx : 0 \u2264 x) : Nat.ceil x \u2264 x + 1 ", "theoremName": "BrunTitchmarsh.Nat.ceil_le_self_add_one", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/BrunTitchmarsh.lean", "module": "PrimeNumberTheoremAnd.BrunTitchmarsh", "jsonFile": "PrimeNumberTheoremAnd.BrunTitchmarsh.jsonl", "positionMetadata": {"lineInFile": 144, "tokenPositionInFile": 4687, 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"Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", 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"Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  trans Nat.floor x + 1\n  \u00b7 norm_cast\n    exact Nat.ceil_le_floor_add_one x\n  gcongr\n  apply Nat.floor_le hx", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 114}}
+{"srcContext": "/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Kontorovich, David Loeffler, Heather Macbeth\n-/\nimport Mathlib.Analysis.Calculus.ParametricIntegral\nimport Mathlib.Analysis.Fourier.AddCircle\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Calculus.FDeriv.Analytic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space\n\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\n\n/-!\n# Derivatives of the Fourier transform\n\nIn this file we compute the Fr\u00e9chet derivative of the Fourier transform of `f`, where `f` is a\nfunction such that both `f` and `v \u21a6 \u2016v\u2016 * \u2016f v\u2016` are integrable. Here the Fourier transform is\nunderstood as an operator `(V \u2192 E) \u2192 (W \u2192 E)`, where `V` and `W` are normed `\u211d`-vector spaces\nand the Fourier transform is taken with respect to a continuous `\u211d`-bilinear\npairing `L : V \u00d7 W \u2192 \u211d` and a given reference measure `\u03bc`.\n\nWe also investigate higher derivatives: Assuming that `\u2016v\u2016^n * \u2016f v\u2016` is integrable, we show\nthat the Fourier transform of `f` is `C^n`.\n\nWe also study in a parallel way the Fourier transform of the derivative, which is obtained by\ntensoring the Fourier transform of the original function with the bilinear form.\n\nWe give specialized versions of these results on inner product spaces (where `L` is the scalar\nproduct) and on the real line, where we express the one-dimensional derivative in more concrete\nterms, as the Fourier transform of `-2\u03c0I x * f x` (or `(-2\u03c0I x)^n * f x` for higher derivatives).\n\n## Main definitions and results\n\nWe introduce two convenience definitions:\n\n* `VectorFourier.fourierSMulRight L f`: given `f : V \u2192 E` and `L` a bilinear pairing\n  between `V` and `W`, then this is the function `fun v \u21a6 -(2 * \u03c0 * I) (L v \u2b1d) \u2022 f v`,\n  from `V` to `Hom (W, E)`.\n  This is essentially `ContinousLinearMap.smulRight`, up to the factor `- 2\u03c0I` designed to make sure\n  that the Fourier integral of `fourierSMulRight L f` is the derivative of the Fourier\n  integral of `f`.\n* `VectorFourier.fourierPowSMulRight` is the higher order analogue for higher derivatives:\n  `fourierPowSMulRight L f v n` is informally `(-(2 * \u03c0 * I))^n (L v \u2b1d)^n \u2022 f v`, in\n  the space of continuous multilinear maps `W [\u00d7n]\u2192L[\u211d] E`.\n\nWith these definitions, the statements read as follows, first in a general context\n(arbitrary `L` and `\u03bc`):\n\n* `VectorFourier.hasFDerivAt_fourierIntegral`: the Fourier integral of `f` is differentiable, with\n    derivative the Fourier integral of `fourierSMulRight L f`.\n* `VectorFourier.differentiable_fourierIntegral`: the Fourier integral of `f` is differentiable.\n* `VectorFourier.fderiv_fourierIntegral`: formula for the derivative of the Fourier integral of `f`.\n* `VectorFourier.fourierIntegral_fderiv`: formula for the Fourier integral of the derivative of `f`.\n* `VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` has an explicit Taylor series up to order `N`, given by the Fourier\n  integrals of `fun v \u21a6 fourierPowSMulRight L f v n`.\n* `VectorFourier.contDiff_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` is `C^n`.\n* `VectorFourier.iteratedFDeriv_fourierIntegral`: under suitable integrability conditions,\n  explicit formula for the `n`-th derivative of the Fourier integral of `f`, as the Fourier\n  integral of `fun v \u21a6 fourierPowSMulRight L f v n`.\n\nThese statements are then specialized to the case of the usual Fourier transform on\nfinite-dimensional inner product spaces with their canonical Lebesgue measure (covering in\nparticular the case of the real line), replacing the namespace `VectorFourier` by\nthe namespace `Real` in the above statements.\n\nWe also give specialized versions of the one-dimensional real derivative (and iterated derivative)\nin `Real.deriv_fourierIntegral` and `Real.iteratedDeriv_fourierIntegral`.\n-/\n\nnoncomputable section\n\nopen Real Complex MeasureTheory Filter TopologicalSpace\n\nopen scoped FourierTransform Topology BigOperators\n\n-- without this local instance, Lean tries first the instance\n-- `secondCountableTopologyEither_of_right` (whose priority is 100) and takes a very long time to\n-- fail. Since we only use the left instance in this file, we make sure it is tried first.\nattribute [local instance 101] secondCountableTopologyEither_of_left\n\nnamespace Real\n\nlemma differentiable_fourierChar : Differentiable \u211d (\ud835\udc1e \u00b7 : \u211d \u2192 \u2102) :=\n  fun x \u21a6 (Real.hasDerivAt_fourierChar x).differentiableAt\n\nlemma deriv_fourierChar (x : \u211d) : deriv (\ud835\udc1e \u00b7 : \u211d \u2192 \u2102) x = 2 * \u03c0 * I * \ud835\udc1e x :=\n  (Real.hasDerivAt_fourierChar x).deriv\n\nvariable {V W : Type*} [NormedAddCommGroup V] [NormedSpace \u211d V]\n  [NormedAddCommGroup W] [NormedSpace \u211d W] (L : V \u2192L[\u211d] W \u2192L[\u211d] \u211d)\n\nlemma hasFDerivAt_fourierChar_neg_bilinear_right (v : V) (w : W) :\n    HasFDerivAt (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102))\n      ((-2 * \u03c0 * I * \ud835\udc1e (-L v w)) \u2022 (ofRealCLM \u2218L (L v))) w := by\n  have ha : HasFDerivAt (fun w' : W \u21a6 L v w') (L v) w := ContinuousLinearMap.hasFDerivAt (L v)\n  convert (hasDerivAt_fourierChar (-L v w)).hasFDerivAt.comp w ha.neg\n  ext y\n  simp only [neg_mul, ContinuousLinearMap.coe_smul', ContinuousLinearMap.coe_comp', Pi.smul_apply,\n    Function.comp_apply, ofRealCLM_apply, smul_eq_mul, ContinuousLinearMap.comp_neg,\n    ContinuousLinearMap.neg_apply, ContinuousLinearMap.smulRight_apply,\n    ContinuousLinearMap.one_apply, real_smul, neg_inj]\n  ring\n\n", "theoremStatement": "lemma fderiv_fourierChar_neg_bilinear_right_apply (v : V) (w y : W) :\n    fderiv \u211d (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102)) w y = -2 * \u03c0 * I * L v y * \ud835\udc1e (-L v w) ", "theoremName": "Real.fderiv_fourierChar_neg_bilinear_right_apply", "fileCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "theoremCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "file": 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"Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", 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"Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "Mathlib.Analysis.Fourier.FourierTransformDeriv"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp [(hasFDerivAt_fourierChar_neg_bilinear_right L v w).fderiv]\n  ring", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 79}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg 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"Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 322}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Logic.Embedding.Basic\nimport Mathlib.NumberTheory.ArithmeticFunction\n\n\n/-!\n# Sets of tuples with a fixed product\nThis file defines the finite set of `d`-tuples of natural numbers with a fixed product `n` as\n`Nat.finMulAntidiagonal`.\n## Main Results\n* There are `d^(\u03c9 n)` ways to write `n` as a product of `d` natural numbers, when `n` is squarefree\n(`card_finMulAntidiagonal_of_squarefree`)\n* There are `3^(\u03c9 n)` pairs of natural numbers whose `lcm` is `n`, when `n` is squarefree\n(`card_pair_lcm_eq`)\n\nSee PR #10668\n-/\n\nopen Finset\nopen scoped BigOperators ArithmeticFunction\n\n\nnamespace Nat\ndef finMulAntidiagonal (d : \u2115) (n : \u2115) : Finset (Fin d \u2192 \u2115) :=\n  aux d n\nwhere\n  /-- Auxiliary construction for `finMulAntidiagonal` that bundles a proof of lawfulness\n  (`mem_finMulAntidiagonal`), as this is needed to invoke `disjiUnion`. Using `Finset.disjiUnion` makes\n  this computationally much more efficient than using `Finset.biUnion`. -/\n  aux (d : \u2115) (n : \u2115) : {s : Finset (Fin d \u2192 \u2115) // \u2200 f, f \u2208 s \u2194 \u220f i, f i = n \u2227 n \u2260 0} :=\n    match d with\n    | 0 =>\n      if h : n = 1 then\n        \u27e8{1}, by simp [h]; exact List.ofFn_inj.mp rfl\u27e9\n      else\n        \u27e8\u2205, by simp [Ne.symm h]\u27e9\n    | d + 1 =>\n      { val := (divisorsAntidiagonal n).disjiUnion\n          (fun ab => (aux d ab.2).1.map {\n              toFun := Fin.cons (ab.1)\n              inj' := Fin.cons_right_injective _ })\n          (fun i _hi j _hj hij => Finset.disjoint_left.2 fun t hti htj => hij <| by\n            simp_rw [Finset.mem_map, Function.Embedding.coeFn_mk] at hti htj\n            obtain \u27e8ai, hai, hij'\u27e9 := hti\n            obtain \u27e8aj, haj, rfl\u27e9 := htj\n            rw [Fin.cons_eq_cons] at hij'\n            ext\n            \u00b7 exact hij'.1\n            \u00b7 obtain \u27e8-, rfl\u27e9 := hij'\n              rw [\u2190 (aux d i.2).prop ai |>.mp hai |>.1, \u2190 (aux d j.2).prop ai |>.mp haj |>.1])\n        property := fun f => by\n          simp_rw [mem_disjiUnion, mem_divisorsAntidiagonal, mem_map, Function.Embedding.coeFn_mk,\n            Prod.exists, (aux d _).prop, Fin.prod_univ_succ]\n          constructor\n          \u00b7 rintro \u27e8a, b, \u27e8rfl, hab\u27e9, g, \u27e8rfl, hb\u27e9, rfl\u27e9\n            simp only [Fin.cons_zero, Fin.cons_succ]\n            exact (true_and_iff (a * \u220f x : Fin d, g x \u2260 0)).mpr hab\n          \u00b7 intro \u27e8rfl, hf\u27e9\n            exact \u27e8_, _, \u27e8rfl, hf\u27e9, _, \u27e8rfl, by exact right_ne_zero_of_mul hf\u27e9, Fin.cons_self_tail f\u27e9 }\n\n@[simp]\ntheorem mem_finMulAntidiagonal {d n : \u2115} {f : (Fin d) \u2192 \u2115} :\n    f \u2208 finMulAntidiagonal d n \u2194 \u220f i, f i = n \u2227 n \u2260 0 :=\n  (finMulAntidiagonal.aux d n).prop f\n\n@[simp]\ntheorem finMulAntidiagonal_zero {d : \u2115} :\n    finMulAntidiagonal d 0 = \u2205 := by\n  ext; simp\n\n", "theoremStatement": "theorem finMulAntidiagonal_one {d : \u2115} :\n    finMulAntidiagonal d 1 = {fun _ => 1} ", "theoremName": "Nat.finMulAntidiagonal_one", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Data/Nat/FinMulAntidiagonal.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal.jsonl", "positionMetadata": {"lineInFile": 76, "tokenPositionInFile": 2804, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 62, "importedModules": ["Init.Prelude", 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"Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.NumberTheory.Divisors", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Data.Rat.BigOperators", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Data.Finsupp.Basic", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.GroupTheory.Finiteness", "Mathlib.RingTheory.Finiteness", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.Data.Nat.Multiplicity", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  ext f; simp only [mem_finMulAntidiagonal, and_true, mem_singleton]\n  constructor\n  \u00b7 intro \u27e8hf, _\u27e9; ext i;\n    rw [\u2190Nat.dvd_one, \u2190hf];\n    exact dvd_prod_of_mem f (mem_univ _)\n  \u00b7 rintro rfl; simp only [prod_const_one, implies_true, ne_eq, one_ne_zero, not_false_eq_true,\n    and_self]", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 293}}
+{"srcContext": "import Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.Analysis.Complex.Convex\n\nopen Complex Set Topology\n\nopen scoped Interval\n\nvariable {z w : \u2102} {c : \u211d}\n\n/-%%\nThis files gathers definitions and basic properties about rectangles.\n%%-/\n\n/-%%\n\\begin{definition}\\label{Rectangle}\\lean{Rectangle}\\leanok\nA Rectangle has corners $z$ and $w \\in \\C$.\n\\end{definition}\n%%-/\n/-- A `Rectangle` has corners `z` and `w`. -/\ndef Rectangle (z w : \u2102) : Set \u2102 := [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]]\n\nnamespace Rectangle\n\nlemma symm : Rectangle z w = Rectangle w z := by\n  simp [Rectangle, uIcc_comm]\n\n", "theoremStatement": "lemma symm_re : Rectangle (w.re + z.im * I) (z.re + w.im * I) = Rectangle z w ", "theoremName": "Rectangle.symm_re", "fileCreated": {"commit": "2b6bc532577556b56d3446de5e18efe7b5a95895", "date": "2024-02-20"}, "theoremCreated": {"commit": "a926b33fd8d4289c58c4efa7b1055cc3cc21e8b5", "date": "2024-02-17"}, 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"Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", 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+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\ntheorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) := by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz\n\ntheorem primesBetween_le_siftedSum_add :\n    primesBetween x (x+y) \u2264 (primeInterSieve x y z hz).siftedSum + z := by\n  classical\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a (Finset.Icc 1 (Nat.floor z))).card\n  \u00b7 rw[primesBetween]\n    norm_cast\n    apply Finset.card_le_card\n    apply primesBetween_subset _ _ _ hx\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card\n    + \u2191(Finset.Icc 1 (Nat.floor z)).card\n  \u00b7 norm_cast\n    apply Finset.card_union_le\n  rw[siftedSum_eq_card]\n  gcongr\n  rw[Nat.card_Icc]\n  simp\n  apply Nat.floor_le\n  linarith\n\nsection Remainder\n\ntheorem Ioc_filter_dvd_eq (d a b: \u2115) (hd : d \u2260 0) :\n  Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b) =\n    Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d)) := by\n  ext n\n  simp only [Finset.mem_filter, Finset.mem_Ioc, Nat.ceil_le, Finset.mem_image,\n    Nat.le_floor_iff (show 0 \u2264 x+y by linarith)]\n  constructor\n  \u00b7 intro hn\n    use  n/d\n    rcases hn with \u27e8\u27e8han, hnb\u27e9, hd\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 exact Nat.div_lt_div_of_lt_of_dvd hd han\n    \u00b7 exact Nat.div_le_div_right (Nat.le_floor hnb)\n    \u00b7 exact Nat.div_mul_cancel hd\n  \u00b7 rintro \u27e8r, \u27e8ha, ha'\u27e9, rfl\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 refine (Nat.div_lt_iff_lt_mul ?_).mp ha\n      omega\n    \u00b7 exact Nat.mul_le_of_le_div d r b ha'\n    \u00b7 exact Nat.dvd_mul_left d r\n\ntheorem card_Ioc_filter_dvd (d a b: \u2115) (hd : d \u2260 0) :\n    (Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b)).card = b / d - a / d  := by\n  rw [Ioc_filter_dvd_eq _ _ _ hd]\n  rw [Finset.card_image_of_injective _ <| mul_left_injective\u2080 hd]\n  simp\n\ntheorem multSum_eq (d : \u2115) (hd : d \u2260 0):\n    (primeInterSieve x y z hz).multSum d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) := by\n  unfold Sieve.multSum\n  rw[primeInterSieve]\n  simp\n  trans \u2191(Finset.Ioc (Nat.ceil x - 1) (Nat.floor (x+y)) |>.filter (d \u2223 \u00b7) |>.card)\n  \u00b7 rw [\u2190Nat.Icc_succ_left]\n    congr\n    rw [\u2190Nat.succ_sub]; rfl\n    simp [hx]\n  \u00b7 rw[BrunTitchmarsh.card_Ioc_filter_dvd _ _ _ hd]\n\ntheorem rem_eq (d : \u2115) (hd : d \u2260 0) : (primeInterSieve x y z hz).rem d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) - (\u2191d)\u207b\u00b9 * y := by\n  unfold Sieve.rem\n  rw[multSum_eq x y z hx hz d hd]\n  simp [primeInterSieve, if_neg hd]\n\ntheorem Nat.ceil_le_self_add_one (x : \u211d) (hx : 0 \u2264 x) : Nat.ceil x \u2264 x + 1 := by\n  trans Nat.floor x + 1\n  \u00b7 norm_cast\n    exact Nat.ceil_le_floor_add_one x\n  gcongr\n  apply Nat.floor_le hx\n\ntheorem floor_approx (x : \u211d) (hx : 0 \u2264 x) : \u2203 C, |C| \u2264 1 \u2227  \u2191((Nat.floor x)) = x + C := by\n  use \u2191(Nat.floor x) - x\n  simp\n  rw[abs_le]\n  constructor\n  \u00b7 simp only [neg_le_sub_iff_le_add]\n    linarith [Nat.lt_floor_add_one x]\n  \u00b7 simp only [tsub_le_iff_right]\n    linarith [Nat.floor_le hx]\n\ntheorem ceil_approx (x : \u211d) (hx : 0 \u2264 x) : \u2203 C, |C| \u2264 1 \u2227  \u2191((Nat.ceil x)) = x + C := by\n  use \u2191(Nat.ceil x) - x\n  simp\n  rw[abs_le]\n  constructor\n  \u00b7 simp only [neg_le_sub_iff_le_add]\n    linarith [Nat.le_ceil x]\n  \u00b7 simp only [tsub_le_iff_right]\n    rw[add_comm]\n    exact Nat.ceil_le_self_add_one x hx\n\ntheorem nat_div_approx (a b : \u2115) : \u2203 C, |C| \u2264 1 \u2227 \u2191(a/b) = (a/b : \u211d) + C := by\n  rw[\u2190Nat.floor_div_eq_div (\u03b1:=\u211d)]\n  apply floor_approx (a/b:\u211d) (by positivity)\n\n", "theoremStatement": "theorem floor_div_approx (x : \u211d) (hx : 0 \u2264 x) (d : \u2115) : \u2203 C, |C| \u2264 2 \u2227  \u2191((Nat.floor x)/d) = x / d + C ", "theoremName": "BrunTitchmarsh.floor_div_approx", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": 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"Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  by_cases hd : d = 0\n  \u00b7 simp [hd]\n  obtain \u27e8C\u2081, hC\u2081_le, hC\u2081\u27e9 := nat_div_approx (Nat.floor x) d\n  obtain \u27e8C\u2082, hC\u2082_le, hC\u2082\u27e9 := floor_approx x hx\n  rw[hC\u2081, hC\u2082]\n  use  C\u2081 + C\u2082/d\n  refine \u27e8?_, by ring\u27e9\n  have : |C\u2081 + C\u2082/d| \u2264 |C\u2081| + |C\u2082/d| := abs_add C\u2081 (C\u2082 / \u2191d)\n  have : |C\u2082/d| \u2264 |C\u2082| := by\n    rw[abs_div]\n    apply div_le_self\n    \u00b7 exact abs_nonneg C\u2082\n    \u00b7 simp only [Nat.abs_cast, Nat.one_le_cast]\n      omega\n  linarith", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 430}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.NumberTheory.PrimeCounting\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Data.Complex.ExponentialBounds\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\n/-!\n# Bounds for the Selberg sieve\nThis file proves a number of results to help bound `Sieve.selbergSum`\n\n## Main Results\n* `selbergBoundingSum_ge_sum_div`: If `\u03bd` is completely multiplicative then `S \u2265 \u2211_{n \u2264 \u221ay}, \u03bd n`\n* `boundingSum_ge_log`: If `\u03bd n = 1 / n` then `S \u2265 log y / 2`\n* `rem_sum_le_of_const`: If `R_d \u2264 C` then the error term is at most `C * y * (1 + log y)^3`\n-/\n\nopen scoped Nat ArithmeticFunction BigOperators Classical\n\nnoncomputable section\nnamespace Sieve\n\nlemma prodDistinctPrimes_squarefree (s : Finset \u2115) (h : \u2200 p \u2208 s, p.Prime) :\n    Squarefree (\u220f p in s, p) := by\n  refine Iff.mpr Nat.squarefree_iff_prime_squarefree ?_\n  intro p hp; by_contra h_dvd\n  by_cases hps : p \u2208 s\n  \u00b7 rw [\u2190Finset.mul_prod_erase (a:=p) (h := hps), mul_dvd_mul_iff_left (Nat.Prime.ne_zero hp)] at h_dvd\n    cases' Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) h_dvd with q hq\n    rw [Finset.mem_erase] at hq\n    exact hq.1.1 $ symm $ (Nat.prime_dvd_prime_iff_eq hp (h q hq.1.2)).mp hq.2\n  \u00b7 have : p \u2223 \u220f p in s, p := Trans.trans (dvd_mul_right p p) h_dvd\n    cases' Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) this with q hq\n    have heq : p = q := by\n      rw [\u2190Nat.prime_dvd_prime_iff_eq hp (h q hq.1)]\n      exact hq.2\n    rw [heq] at hps; exact hps hq.1\n\nlemma primorial_squarefree (n : \u2115) : Squarefree (primorial n) := by\n  apply prodDistinctPrimes_squarefree\n  simp_rw [Finset.mem_filter];\n  exact fun _ h => h.2\n\ntheorem zeta_pos_of_prime : \u2200 (p : \u2115), Nat.Prime p \u2192 (0:\u211d) < (\u2191\u03b6:ArithmeticFunction \u211d) p := by\n  intro p hp\n  rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, if_neg (Nat.Prime.ne_zero hp)]\n  norm_num\n\ntheorem zeta_lt_self_of_prime : \u2200 (p : \u2115), Nat.Prime p \u2192 (\u2191\u03b6:ArithmeticFunction \u211d) p < (p:\u211d) := by\n  intro p hp\n  rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, if_neg (Nat.Prime.ne_zero hp)]\n  norm_num;\n  exact Nat.succ_le.mp (Nat.Prime.two_le hp)\n\ntheorem prime_dvd_primorial_iff (n p : \u2115) (hp : p.Prime) :\n    p \u2223 primorial n \u2194 p \u2264 n := by\n  unfold primorial\n  constructor\n  \u00b7 intro h\n    let h' : \u2203 i, i \u2208 Finset.filter Nat.Prime (Finset.range (n + 1)) \u2227 p \u2223 i := Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) h\n    cases' h' with q hq\n    rw [Finset.mem_filter, Finset.mem_range] at hq\n    rw [prime_dvd_prime_iff_eq (Nat.Prime.prime hp) (Nat.Prime.prime hq.1.2)] at hq\n    rw [hq.2]\n    exact Nat.lt_succ.mp hq.1.1\n  \u00b7 intro h\n    apply Finset.dvd_prod_of_mem\n    rw [Finset.mem_filter, Finset.mem_range]\n    exact \u27e8Nat.lt_succ.mpr h, hp\u27e9\n\ntheorem siftedSum_eq (s : SelbergSieve) (hw : \u2200 i \u2208 s.support, s.weights i = 1) (z : \u211d) (hz : 1 \u2264 z) (hP : s.prodPrimes = primorial (Nat.floor z)) :\n    s.siftedSum = (s.support.filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  dsimp only [Sieve.siftedSum]\n  rw [Finset.card_eq_sum_ones, \u2190Finset.sum_filter, Nat.cast_sum]\n  apply Finset.sum_congr;\n  rw [hP]\n  ext d; constructor\n  \u00b7 intro hd\n    rw [Finset.mem_filter] at *\n    constructor\n    \u00b7 exact hd.1\n    \u00b7 intro p hpp hpy\n      rw [\u2190Nat.Prime.coprime_iff_not_dvd hpp]\n      apply Nat.Coprime.coprime_dvd_left _ hd.2\n      rw [prime_dvd_primorial_iff _ _ hpp]\n      apply Nat.le_floor hpy\n  \u00b7 intro h\n    rw [Finset.mem_filter] at *\n    constructor\n    \u00b7 exact h.1\n    refine Nat.coprime_of_dvd ?_\n    intro p hp\n    erw [prime_dvd_primorial_iff _ _ hp]\n    intro hpy\n    apply h.2 p hp\n    trans \u2191(Nat.floor z)\n    \u00b7 norm_cast\n    \u00b7 apply Nat.floor_le\n      linarith only [hz]\n  simp_rw [Nat.cast_one]\n  intro x hx\n  simp only [Finset.filter_congr_decidable, Finset.mem_filter] at hx\n  apply hw x hx.1\n\ndef CompletelyMultiplicative (f : ArithmeticFunction \u211d) : Prop := f 1 = 1 \u2227 \u2200 a b, f (a*b) = f a * f b\n\nnamespace CompletelyMultiplicative\nopen ArithmeticFunction\ntheorem zeta : CompletelyMultiplicative \u03b6 := by\n  unfold CompletelyMultiplicative\n  simp_rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, ite_false, Nat.cast_one,\n    mul_eq_zero, Nat.cast_ite, CharP.cast_eq_zero, mul_ite, mul_zero, true_and]\n  intro a b;\n  by_cases ha : a = 0\n  \u00b7 simp [ha]\n  by_cases hb : b = 0\n  \u00b7 simp [hb]\n  rw [if_neg, if_neg hb, if_neg ha]; ring\n  push_neg; exact \u27e8ha, hb\u27e9\n\ntheorem id : CompletelyMultiplicative ArithmeticFunction.id := by\n    constructor <;> simp\n\n", "theoremStatement": "theorem pmul (f g : ArithmeticFunction \u211d) (hf : CompletelyMultiplicative f) (hg : CompletelyMultiplicative g) :\n    CompletelyMultiplicative (ArithmeticFunction.pmul f g) ", "theoremName": "Sieve.CompletelyMultiplicative.pmul", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "44fca7b36e6e3de82ce0278311d181bdf7d70f4c", "date": "2024-01-12"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/SelbergBounds.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds.jsonl", "positionMetadata": {"lineInFile": 129, "tokenPositionInFile": 4716, "theoremPositionInFile": 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"Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Prime", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Int.ModEq", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  constructor\n  \u00b7 rw [pmul_apply, hf.1, hg.1, mul_one]\n  intro a b\n  simp_rw [pmul_apply, hf.2, hg.2]; ring", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 113}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\nlemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]\n\nlemma sum_eq_int_deriv_aux_lt {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_lt_kpOne : b < k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k := Int.floor_eq_iff.mpr \u27e8by linarith [ha.1, ha.2], by linarith\u27e9\n  simp only [flb_eq_k, gt_iff_lt, lt_add_iff_pos_right, zero_lt_one, Finset.Icc_eq_empty_of_lt,\n    Finset.sum_empty]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont]\n  have : Finset.Ioc k k = {} := by simp only [ge_iff_le, le_refl, Finset.Ioc_eq_empty_of_le]\n  simp only [this, Finset.sum_empty, one_div]; ring_nf\n\nlemma sum_eq_int_deriv_aux1 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  by_cases h : b = k + 1\n  \u00b7 exact sum_eq_int_deriv_aux_eq h \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact sum_eq_int_deriv_aux_lt ha (Ne.lt_of_le h b_le_kpOne) \u03c6Diff deriv\u03c6Cont\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv_aux]\\label{sum_eq_int_deriv_aux}\\lean{sum_eq_int_deriv_aux}\\leanok\n  Let $k \\le a < b\\le k+1$, with $k$ an integer, and let $\\phi$ be continuously differentiable on\n  $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma sum_eq_int_deriv_aux {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  have fl_a_eq_k : \u230aa\u230b = k := Int.floor_eq_iff.mpr \u27e8ha.1, by linarith [ha.2]\u27e9\n  convert sum_eq_int_deriv_aux1 ha b_le_kpOne \u03c6Diff deriv\u03c6Cont using 2\n  \u00b7 rw [fl_a_eq_k]\n  \u00b7 congr\n  \u00b7 apply intervalIntegral.integral_congr_ae\n    have : \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.volume, x \u2260 b := by\n      convert Countable.ae_not_mem (s := {b}) (by simp) (\u03bc := MeasureTheory.volume) using 1\n    filter_upwards [this]\n    intro x x_ne_b hx\n    rw [uIoc_of_le ha.2.le, mem_Ioc] at hx\n    congr\n    exact Int.floor_eq_iff.mpr \u27e8by linarith [ha.1], by have := Ne.lt_of_le x_ne_b hx.2; linarith\u27e9\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\n-- Thanks to Arend Mellendijk\n\nlemma interval_induction_aux_int (n : \u2115) : \u2200 (P : \u211d \u2192 \u211d \u2192 Prop)\n    (_ : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (_ : \u2200 (a : \u211d) (k : \u2124) (c : \u211d), a < k \u2192 k < c \u2192 P a k \u2192 P k c \u2192 P a c)\n    (a b : \u211d) (_ : a < b) (_ : n = \u230ab\u230b - \u230aa\u230b),\n    P a b := by\n  induction n using Nat.case_strong_induction_on with\n  | hz =>\n    intro P base _ a b hab hn\n    apply base a b \u230aa\u230b (Int.floor_le a) hab\n    rw [(by simp only [CharP.cast_eq_zero] at hn; linarith : \u230aa\u230b = \u230ab\u230b)]\n    exact (Int.lt_floor_add_one b).le\n  | hi n ih =>\n    intro P base step a b _ hn\n    have Pa : P a (\u230aa\u230b + 1) :=\n      base a (\u230aa\u230b + 1) \u230aa\u230b (Int.floor_le a) (Int.lt_floor_add_one a) (le_of_eq rfl)\n    by_cases b_le_flaP1 : b = \u230aa\u230b + 1\n    \u00b7 rwa [b_le_flaP1]\n    have flaP1_lt_b : \u230aa\u230b + 1 < b := by\n      simp only [Nat.cast_succ] at hn\n      have : (\u230aa\u230b : \u211d) + 1 \u2264 \u230ab\u230b := by exact_mod_cast (by linarith)\n      exact Ne.lt_of_le (id (Ne.symm b_le_flaP1)) (by linarith [Int.floor_le b] : \u230aa\u230b + 1 \u2264 b)\n    have Pfla_b : P (\u230aa\u230b + 1) b := by\n      apply ih n (le_of_eq rfl) P base step (\u230aa\u230b + 1) b flaP1_lt_b\n      simp only [Int.floor_add_one, Int.floor_intCast, Nat.cast_succ] at hn \u22a2\n      linarith\n    refine step a (\u230aa\u230b + 1) b ?_ (by exact_mod_cast flaP1_lt_b) (by exact_mod_cast Pa)\n      (by exact_mod_cast Pfla_b)\n    have := Int.lt_floor_add_one a\n    exact_mod_cast this\n\n", "theoremStatement": "lemma interval_induction (P : \u211d \u2192 \u211d \u2192 Prop)\n    (base : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (step : \u2200 (a : \u211d) (k : \u2124) (b : \u211d), a < k \u2192 k < b \u2192 P a k \u2192 P k b \u2192 P a b)\n    (a b : \u211d) (hab : a < b) : P a b ", "theoremName": "interval_induction", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "8e5c683f6945de3190aa88ae5e78a1eff695c2a2", "date": "2024-03-28"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": 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"Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  set n := \u230ab\u230b - \u230aa\u230b with hn\n  clear_value n\n  have : 0 \u2264 n := by simp only [hn, sub_nonneg, ge_iff_le, Int.floor_le_floor _ _ (hab.le)]\n  lift n to \u2115 using this\n  exact interval_induction_aux_int n P base step a b hab hn", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 227}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\n", "theoremStatement": "lemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n ", "theoremName": "cumsum_succ", "fileCreated": 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"Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp [cumsum, Finset.sum_range_succ]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 44}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\nlemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self\n\nlemma bound_sum_log0 {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let f0 i := if i = 0 then 0 else f i\n  have l1 : chebyWith C f0 := by\n    intro n ; refine Finset.sum_le_sum (fun i _ => ?_) |>.trans (hf n)\n    by_cases hi : i = 0 <;> simp [hi, f0]\n  have l2 i : \u2016f i\u2016 / i = \u2016f0 i\u2016 / i := by by_cases hi : i = 0 <;> simp [hi, f0]\n  simp_rw [l2] ; apply bound_sum_log rfl l1 hx\n\nlemma bound_sum_log' {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + 2 * \u03c0 ^ 2) := by\n  simpa only [hh_integral'] using bound_sum_log0 hf hx\n\nlemma summable_fourier (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  exact Summable.of_nonneg_of_le (fun _ => norm_nonneg _) l6 (by simpa using l5.const_smul (W21.norm \u03c8))\n\nlemma bound_I1 (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264\n    W21.norm \u03c8 \u2022 \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 := by\n\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  have l1 : Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n    exact summable_fourier x hx \u03c8 hcheby\n  apply (norm_tsum_le_tsum_norm l1).trans\n  simpa only [\u2190 tsum_const_smul _ l5] using tsum_mono l1 (by simpa using l5.const_smul (W21.norm \u03c8)) l6\n\nlemma bound_I1' {C : \u211d} (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21) (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264 W21.norm \u03c8 * C * (1 + 2 * \u03c0 ^ 2) := by\n\n  apply bound_I1 x (by linarith) \u03c8 \u27e8_, hcheby\u27e9 |>.trans\n  rw [smul_eq_mul, mul_assoc]\n  apply mul_le_mul le_rfl (bound_sum_log' hcheby hx) ?_ W21.norm_nonneg\n  apply tsum_nonneg (fun i => by positivity)\n\nlemma bound_I2 (x : \u211d) (\u03c8 : W21) :\n    \u2016\u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (2 * \u03c0 ^ 2) := by\n\n  have key a : \u2016\ud835\udcd5 \u03c8 (a / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := decay_bounds_key \u03c8 _\n  have twopi : 0 \u2264 2 * \u03c0 := by simp [pi_nonneg]\n  have l3 : Integrable (fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.comp_div (by norm_num [pi_ne_zero])\n  have l2 : IntegrableOn (fun i \u21a6 W21.norm \u03c8 * (1 + (i / (2 * \u03c0)) ^ 2)\u207b\u00b9) (Ici (-Real.log x)) := by\n    exact (l3.const_mul _).integrableOn\n  have l1 : IntegrableOn (fun i \u21a6 \u2016\ud835\udcd5 \u03c8 (i / (2 * \u03c0))\u2016) (Ici (-Real.log x)) := by\n    refine ((l3.const_mul (W21.norm \u03c8)).mono' ?_ ?_).integrableOn\n    \u00b7 apply Continuous.aestronglyMeasurable ; continuity\n    \u00b7 simp only [norm_norm, key] ; simp\n  have l5 : 0 \u2264\u1d50[volume] fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := by apply eventually_of_forall ; intro x ; positivity\n  refine (norm_integral_le_integral_norm _).trans <| (set_integral_mono l1 l2 key).trans ?_\n  rw [integral_mul_left] ; gcongr ; apply W21.norm_nonneg\n  refine (set_integral_le_integral l3 l5).trans ?_\n  rw [Measure.integral_comp_div (fun x => (1 + x ^ 2)\u207b\u00b9) (2 * \u03c0)]\n  simp [abs_eq_self.mpr twopi] ; ring_nf ; rfl\n\nlemma bound_main {C : \u211d} (A : \u2102) (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21)\n    (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264\n      W21.norm \u03c8 * (C * (1 + 2 * \u03c0 ^ 2) + \u2016A\u2016 * (2 * \u03c0 ^ 2)) := by\n\n  have l1 := bound_I1' x hx \u03c8 hcheby\n  have l2 := mul_le_mul (le_refl \u2016A\u2016) (bound_I2 x \u03c8) (by positivity) (by positivity)\n  apply norm_sub_le _ _ |>.trans ; rw [norm_mul]\n  convert _root_.add_le_add l1 l2 using 1 ; ring\n\n/-%%\n\\begin{lemma}[Limiting identity for Schwartz functions]\\label{schwarz-id}\\lean{limiting_cor_schwartz}\\leanok  The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.\n\\end{lemma}\n%%-/\n\n", "theoremStatement": "lemma limiting_cor_W21 (\u03c8 : W21) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) ", "theoremName": "limiting_cor_W21", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "d5b16ca8ce6fad2292b07f2cfc5fd60662cbdcca", "date": "2024-03-29"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1504, "tokenPositionInFile": 74226, "theoremPositionInFile": 128}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 8, "repositoryPremises": true, "numRepositoryPremises": 22, "numPremises": 501, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", 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"Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  -- Shorter notation for clarity\n  let S1 x (\u03c8 : \u211d \u2192 \u2102) := \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191n / x))\n  let S2 x (\u03c8 : \u211d \u2192 \u2102) := \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\n  let S x \u03c8 := S1 x \u03c8 - S2 x \u03c8 ; change Tendsto (fun x \u21a6 S x \u03c8) atTop (\ud835\udcdd 0)\n\n  -- Build the truncation\n  obtain g := exists_trunc\n  let \u03a8 R := g.scale R * \u03c8\n  have key R : Tendsto (fun x \u21a6 S x (\u03a8 R)) atTop (\ud835\udcdd 0) := limiting_cor (\u03a8 R) hf hcheby hG hG'\n\n  -- Choose the truncation radius\n  obtain \u27e8C, hcheby\u27e9 := hcheby\n  have hC : 0 \u2264 C := by\n    have : \u2016f 0\u2016 \u2264 C := by simpa [cumsum] using hcheby 1\n    have : 0 \u2264 \u2016f 0\u2016 := by positivity\n    linarith\n  have key2 : Tendsto (fun R \u21a6 W21.norm (\u03c8 - \u03a8 R)) atTop (\ud835\udcdd 0) := W21_approximation \u03c8 g\n  simp_rw [Metric.tendsto_nhds] at key key2 \u22a2 ; intro \u03b5 h\u03b5\n  let M := C * (1 + 2 * \u03c0 ^ 2) + \u2016(A : \u2102)\u2016 * (2 * \u03c0 ^ 2)\n  obtain \u27e8R, hR\u03c8\u27e9 := (key2 ((\u03b5 / 2) / (1 + M)) (by positivity)).exists\n  simp only [dist_zero_right, Real.norm_eq_abs, abs_eq_self.mpr W21.norm_nonneg] at hR\u03c8 key\n\n  -- Apply the compact support case\n  filter_upwards [eventually_ge_atTop 1, key R (\u03b5 / 2) (by positivity)] with x hx key\n\n  -- Control the tail term\n  have key3 : \u2016S x (\u03c8 - \u03a8 R)\u2016 < \u03b5 / 2 := by\n    have : \u2016S x _\u2016 \u2264 _ * M := @bound_main f C A x hx (\u03c8 - \u03a8 R) hcheby\n    apply this.trans_lt\n    apply (mul_le_mul (d := 1 + M) le_rfl (by simp) (by positivity) W21.norm_nonneg).trans_lt\n    have : 0 < 1 + M := by positivity\n    convert (mul_lt_mul_right this).mpr hR\u03c8 using 1 ; field_simp ; ring\n\n  -- Conclude the proof\n  have S1_sub_1 x : \ud835\udcd5 (\u21d1\u03c8 - \u21d1(\u03a8 R)) x = \ud835\udcd5 \u03c8 x - \ud835\udcd5 (\u03a8 R) x := by\n    have l1 : AEStronglyMeasurable (fun x_1 : \u211d \u21a6 cexp (-(2 * \u2191\u03c0 * (\u2191x_1 * \u2191x) * I))) volume := by\n      refine (Continuous.mul ?_ continuous_const).neg.cexp.aestronglyMeasurable\n      apply continuous_const.mul <| contDiff_ofReal.continuous.mul continuous_const\n    simp [Real.fourierIntegral_eq', mul_sub] ; apply integral_sub\n    \u00b7 apply \u03c8.hf.bdd_mul l1 ; use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n    \u00b7 apply (\u03a8 R : W21) |>.hf |>.bdd_mul l1\n      use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n\n  have S1_sub : S1 x (\u03c8 - \u03a8 R) = S1 x \u03c8 - S1 x (\u03a8 R) := by\n    simp [S1, S1_sub_1, mul_sub] ; apply tsum_sub\n    \u00b7 have := summable_fourier x (by positivity) \u03c8 \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n    \u00b7 have := summable_fourier x (by positivity) (\u03a8 R) \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n\n  have S2_sub : S2 x (\u03c8 - \u03a8 R) = S2 x \u03c8 - S2 x (\u03a8 R) := by\n    simp [S2, S1_sub_1] ; rw [integral_sub] ; ring\n    \u00b7 exact \u03c8.integrable_fourier (by positivity) |>.restrict\n    \u00b7 exact (\u03a8 R : W21).integrable_fourier (by positivity) |>.restrict\n\n  have S_sub : S x (\u03c8 - \u03a8 R) = S x \u03c8 - S x (\u03a8 R) := by simp [S, S1_sub, S2_sub] ; ring\n  simpa [S_sub, \u03a8] using norm_add_le _ _ |>.trans_lt (_root_.add_lt_add key3 key)", "proofType": "tactic", "proofLengthLines": 60, "proofLengthTokens": 2892}}
+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\ntheorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) := by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz\n\ntheorem primesBetween_le_siftedSum_add :\n    primesBetween x (x+y) \u2264 (primeInterSieve x y z hz).siftedSum + z := by\n  classical\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a (Finset.Icc 1 (Nat.floor z))).card\n  \u00b7 rw[primesBetween]\n    norm_cast\n    apply Finset.card_le_card\n    apply primesBetween_subset _ _ _ hx\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card\n    + \u2191(Finset.Icc 1 (Nat.floor z)).card\n  \u00b7 norm_cast\n    apply Finset.card_union_le\n  rw[siftedSum_eq_card]\n  gcongr\n  rw[Nat.card_Icc]\n  simp\n  apply Nat.floor_le\n  linarith\n\nsection Remainder\n\ntheorem Ioc_filter_dvd_eq (d a b: \u2115) (hd : d \u2260 0) :\n  Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b) =\n    Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d)) := by\n  ext n\n  simp only [Finset.mem_filter, Finset.mem_Ioc, Nat.ceil_le, Finset.mem_image,\n    Nat.le_floor_iff (show 0 \u2264 x+y by linarith)]\n  constructor\n  \u00b7 intro hn\n    use  n/d\n    rcases hn with \u27e8\u27e8han, hnb\u27e9, hd\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 exact Nat.div_lt_div_of_lt_of_dvd hd han\n    \u00b7 exact Nat.div_le_div_right (Nat.le_floor hnb)\n    \u00b7 exact Nat.div_mul_cancel hd\n  \u00b7 rintro \u27e8r, \u27e8ha, ha'\u27e9, rfl\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 refine (Nat.div_lt_iff_lt_mul ?_).mp ha\n      omega\n    \u00b7 exact Nat.mul_le_of_le_div d r b ha'\n    \u00b7 exact Nat.dvd_mul_left d r\n\ntheorem card_Ioc_filter_dvd (d a b: \u2115) (hd : d \u2260 0) :\n    (Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b)).card = b / d - a / d  := by\n  rw [Ioc_filter_dvd_eq _ _ _ hd]\n  rw [Finset.card_image_of_injective _ <| mul_left_injective\u2080 hd]\n  simp\n\ntheorem multSum_eq (d : \u2115) (hd : d \u2260 0):\n    (primeInterSieve x y z hz).multSum d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) := by\n  unfold Sieve.multSum\n  rw[primeInterSieve]\n  simp\n  trans \u2191(Finset.Ioc (Nat.ceil x - 1) (Nat.floor (x+y)) |>.filter (d \u2223 \u00b7) |>.card)\n  \u00b7 rw [\u2190Nat.Icc_succ_left]\n    congr\n    rw [\u2190Nat.succ_sub]; rfl\n    simp [hx]\n  \u00b7 rw[BrunTitchmarsh.card_Ioc_filter_dvd _ _ _ hd]\n\ntheorem rem_eq (d : \u2115) (hd : d \u2260 0) : (primeInterSieve x y z hz).rem d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) - (\u2191d)\u207b\u00b9 * y := by\n  unfold Sieve.rem\n  rw[multSum_eq x y z hx hz d hd]\n  simp [primeInterSieve, if_neg hd]\n\ntheorem Nat.ceil_le_self_add_one (x : \u211d) (hx : 0 \u2264 x) : Nat.ceil x \u2264 x + 1 := by\n  trans Nat.floor x + 1\n  \u00b7 norm_cast\n    exact Nat.ceil_le_floor_add_one x\n  gcongr\n  apply Nat.floor_le hx\n\ntheorem floor_approx (x : \u211d) (hx : 0 \u2264 x) : \u2203 C, |C| \u2264 1 \u2227  \u2191((Nat.floor x)) = x + C := by\n  use \u2191(Nat.floor x) - x\n  simp\n  rw[abs_le]\n  constructor\n  \u00b7 simp only [neg_le_sub_iff_le_add]\n    linarith [Nat.lt_floor_add_one x]\n  \u00b7 simp only [tsub_le_iff_right]\n    linarith [Nat.floor_le hx]\n\ntheorem ceil_approx (x : \u211d) (hx : 0 \u2264 x) : \u2203 C, |C| \u2264 1 \u2227  \u2191((Nat.ceil x)) = x + C := by\n  use \u2191(Nat.ceil x) - x\n  simp\n  rw[abs_le]\n  constructor\n  \u00b7 simp only [neg_le_sub_iff_le_add]\n    linarith [Nat.le_ceil x]\n  \u00b7 simp only [tsub_le_iff_right]\n    rw[add_comm]\n    exact Nat.ceil_le_self_add_one x hx\n\ntheorem nat_div_approx (a b : \u2115) : \u2203 C, |C| \u2264 1 \u2227 \u2191(a/b) = (a/b : \u211d) + C := by\n  rw[\u2190Nat.floor_div_eq_div (\u03b1:=\u211d)]\n  apply floor_approx (a/b:\u211d) (by positivity)\n\ntheorem floor_div_approx (x : \u211d) (hx : 0 \u2264 x) (d : \u2115) : \u2203 C, |C| \u2264 2 \u2227  \u2191((Nat.floor x)/d) = x / d + C := by\n  by_cases hd : d = 0\n  \u00b7 simp [hd]\n  obtain \u27e8C\u2081, hC\u2081_le, hC\u2081\u27e9 := nat_div_approx (Nat.floor x) d\n  obtain \u27e8C\u2082, hC\u2082_le, hC\u2082\u27e9 := floor_approx x hx\n  rw[hC\u2081, hC\u2082]\n  use  C\u2081 + C\u2082/d\n  refine \u27e8?_, by ring\u27e9\n  have : |C\u2081 + C\u2082/d| \u2264 |C\u2081| + |C\u2082/d| := abs_add C\u2081 (C\u2082 / \u2191d)\n  have : |C\u2082/d| \u2264 |C\u2082| := by\n    rw[abs_div]\n    apply div_le_self\n    \u00b7 exact abs_nonneg C\u2082\n    \u00b7 simp only [Nat.abs_cast, Nat.one_le_cast]\n      omega\n  linarith\n\ntheorem abs_rem_le {d : \u2115} (hd : d \u2260 0) : |(primeInterSieve x y z hz).rem d| \u2264 5 := by\n  rw[rem_eq _ _ _ hx hz _ hd]\n\n  have hpush : \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) = ( \u2191(\u230ax + y\u230b\u208a / d) - \u2191((\u2308x\u2309\u208a - 1) / d) : \u211d) := by\n    rw [Nat.cast_sub]\n    gcongr\n    rw[Nat.le_floor_iff]\n    rw[\u2190add_le_add_iff_right 1]\n    norm_cast\n    rw [Nat.sub_add_cancel (by simp [hx])]\n    linarith [Nat.ceil_le_self_add_one x (le_of_lt hx)]\n    linarith\n\n  rw[hpush]\n  obtain \u27e8C\u2081, hC\u2081_le, hC\u2081\u27e9 := floor_div_approx (x + y) (by linarith) d\n  obtain \u27e8C\u2082, hC\u2082_le, hC\u2082\u27e9 := nat_div_approx (Nat.ceil x - 1) d\n  obtain \u27e8C\u2083, hC\u2083_le, hC\u2083\u27e9 := ceil_approx (x) (by linarith)\n  rw[hC\u2081, hC\u2082, Nat.cast_sub, hC\u2083]\n  ring_nf\n  have : |(\u2191d)\u207b\u00b9 - (\u2191d)\u207b\u00b9 * C\u2083 + (C\u2081 - C\u2082)| \u2264 |(\u2191d)\u207b\u00b9 - (\u2191d)\u207b\u00b9*C\u2083| + |C\u2081 - C\u2082| := by\n    apply (abs_add _ _)\n  have : |(\u2191d)\u207b\u00b9 - (\u2191d)\u207b\u00b9*C\u2083| \u2264 |(\u2191d)\u207b\u00b9| + |(\u2191d)\u207b\u00b9*C\u2083| := abs_sub _ _\n  have : |C\u2081 - C\u2082| \u2264 |C\u2081| + |C\u2082| := abs_sub _ _\n  have : |(d:\u211d)\u207b\u00b9| \u2264 1 := by\n    rw[abs_inv]\n    simp only [Nat.abs_cast]\n    apply Nat.cast_inv_le_one\n  have : |(\u2191d)\u207b\u00b9*C\u2083| \u2264 |C\u2083| := by\n    rw[inv_mul_eq_div, abs_div]\n    apply div_le_self\n    \u00b7 exact abs_nonneg _\n    \u00b7 simp only [Nat.abs_cast, Nat.one_le_cast]\n      omega\n  linarith\n  \u00b7 simp [hx]\n\nend Remainder\n\n", "theoremStatement": "theorem boudingSum_ge : (primeInterSieve x y z hz).selbergBoundingSum \u2265 Real.log z / 2 ", "theoremName": "BrunTitchmarsh.boudingSum_ge", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/BrunTitchmarsh.lean", "module": "PrimeNumberTheoremAnd.BrunTitchmarsh", "jsonFile": "PrimeNumberTheoremAnd.BrunTitchmarsh.jsonl", "positionMetadata": {"lineInFile": 231, "tokenPositionInFile": 7401, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 46, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", 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"Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply boundingSum_ge_log\n  \u00b7 rfl\n  \u00b7 intro p hpp hp\n    erw [prime_dvd_primorial_iff]\n    apply Nat.le_floor\n    exact hp\n    exact hpp", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 143}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\nlemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]\n\nlemma sum_eq_int_deriv_aux_lt {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_lt_kpOne : b < k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k := Int.floor_eq_iff.mpr \u27e8by linarith [ha.1, ha.2], by linarith\u27e9\n  simp only [flb_eq_k, gt_iff_lt, lt_add_iff_pos_right, zero_lt_one, Finset.Icc_eq_empty_of_lt,\n    Finset.sum_empty]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont]\n  have : Finset.Ioc k k = {} := by simp only [ge_iff_le, le_refl, Finset.Ioc_eq_empty_of_le]\n  simp only [this, Finset.sum_empty, one_div]; ring_nf\n\nlemma sum_eq_int_deriv_aux1 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  by_cases h : b = k + 1\n  \u00b7 exact sum_eq_int_deriv_aux_eq h \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact sum_eq_int_deriv_aux_lt ha (Ne.lt_of_le h b_le_kpOne) \u03c6Diff deriv\u03c6Cont\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv_aux]\\label{sum_eq_int_deriv_aux}\\lean{sum_eq_int_deriv_aux}\\leanok\n  Let $k \\le a < b\\le k+1$, with $k$ an integer, and let $\\phi$ be continuously differentiable on\n  $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma sum_eq_int_deriv_aux {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  have fl_a_eq_k : \u230aa\u230b = k := Int.floor_eq_iff.mpr \u27e8ha.1, by linarith [ha.2]\u27e9\n  convert sum_eq_int_deriv_aux1 ha b_le_kpOne \u03c6Diff deriv\u03c6Cont using 2\n  \u00b7 rw [fl_a_eq_k]\n  \u00b7 congr\n  \u00b7 apply intervalIntegral.integral_congr_ae\n    have : \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.volume, x \u2260 b := by\n      convert Countable.ae_not_mem (s := {b}) (by simp) (\u03bc := MeasureTheory.volume) using 1\n    filter_upwards [this]\n    intro x x_ne_b hx\n    rw [uIoc_of_le ha.2.le, mem_Ioc] at hx\n    congr\n    exact Int.floor_eq_iff.mpr \u27e8by linarith [ha.1], by have := Ne.lt_of_le x_ne_b hx.2; linarith\u27e9\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\n-- Thanks to Arend Mellendijk\n\nlemma interval_induction_aux_int (n : \u2115) : \u2200 (P : \u211d \u2192 \u211d \u2192 Prop)\n    (_ : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (_ : \u2200 (a : \u211d) (k : \u2124) (c : \u211d), a < k \u2192 k < c \u2192 P a k \u2192 P k c \u2192 P a c)\n    (a b : \u211d) (_ : a < b) (_ : n = \u230ab\u230b - \u230aa\u230b),\n    P a b := by\n  induction n using Nat.case_strong_induction_on with\n  | hz =>\n    intro P base _ a b hab hn\n    apply base a b \u230aa\u230b (Int.floor_le a) hab\n    rw [(by simp only [CharP.cast_eq_zero] at hn; linarith : \u230aa\u230b = \u230ab\u230b)]\n    exact (Int.lt_floor_add_one b).le\n  | hi n ih =>\n    intro P base step a b _ hn\n    have Pa : P a (\u230aa\u230b + 1) :=\n      base a (\u230aa\u230b + 1) \u230aa\u230b (Int.floor_le a) (Int.lt_floor_add_one a) (le_of_eq rfl)\n    by_cases b_le_flaP1 : b = \u230aa\u230b + 1\n    \u00b7 rwa [b_le_flaP1]\n    have flaP1_lt_b : \u230aa\u230b + 1 < b := by\n      simp only [Nat.cast_succ] at hn\n      have : (\u230aa\u230b : \u211d) + 1 \u2264 \u230ab\u230b := by exact_mod_cast (by linarith)\n      exact Ne.lt_of_le (id (Ne.symm b_le_flaP1)) (by linarith [Int.floor_le b] : \u230aa\u230b + 1 \u2264 b)\n    have Pfla_b : P (\u230aa\u230b + 1) b := by\n      apply ih n (le_of_eq rfl) P base step (\u230aa\u230b + 1) b flaP1_lt_b\n      simp only [Int.floor_add_one, Int.floor_intCast, Nat.cast_succ] at hn \u22a2\n      linarith\n    refine step a (\u230aa\u230b + 1) b ?_ (by exact_mod_cast flaP1_lt_b) (by exact_mod_cast Pa)\n      (by exact_mod_cast Pfla_b)\n    have := Int.lt_floor_add_one a\n    exact_mod_cast this\n\nlemma interval_induction (P : \u211d \u2192 \u211d \u2192 Prop)\n    (base : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (step : \u2200 (a : \u211d) (k : \u2124) (b : \u211d), a < k \u2192 k < b \u2192 P a k \u2192 P k b \u2192 P a b)\n    (a b : \u211d) (hab : a < b) : P a b := by\n  set n := \u230ab\u230b - \u230aa\u230b with hn\n  clear_value n\n  have : 0 \u2264 n := by simp only [hn, sub_nonneg, ge_iff_le, Int.floor_le_floor _ _ (hab.le)]\n  lift n to \u2115 using this\n  exact interval_induction_aux_int n P base step a b hab hn\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv]\\label{sum_eq_int_deriv}\\lean{sum_eq_int_deriv}\\leanok\n  Let $a < b$, and let $\\phi$ be continuously differentiable on $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\n/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {\u03b1 : Type*} [LinearOrder \u03b1] [LocallyFiniteOrder \u03b1]\n    {a b c: \u03b1} [DecidableEq \u03b1] (hb : b \u2208 Icc a c) : Ioc a b = Ioc a c \\ Ioc b c := by\n  ext x\n  simp only [mem_Ioc, mem_sdiff, not_and, not_le]\n  constructor\n  \u00b7 refine fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8\u27e8h\u2081, le_trans h\u2082 (mem_Icc.mp hb).2\u27e9, by contrapose! h\u2082; exact h\u2082.1\u27e9\n  \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8h\u2081.1, by contrapose! h\u2082; exact \u27e8h\u2082, h\u2081.2\u27e9\u27e9\n\n-- In Ya\u00ebl Dillies's API (https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Esum_add_adjacent_intervals/near/430127101)\nlemma Finset.sum_Ioc_add_sum_Ioc {a b c : \u2124} (f : \u2124 \u2192 \u2102) (hb : b \u2208 Icc a c):\n    (\u2211 n in Finset.Ioc a b, f n) + (\u2211 n in Finset.Ioc b c, f n) = \u2211 n in Finset.Ioc a c, f n := by\n  convert Finset.sum_sdiff (s\u2081 := Finset.Ioc b c) (s\u2082 := Finset.Ioc a c) ?_\n  \u00b7 exact Finset.Ioc_diff_Ioc hb\n  \u00b7 exact Finset.Ioc_subset_Ioc (mem_Icc.mp hb).1 (by rfl)\n\nlemma integrability_aux\u2080 {a b : \u211d} :\n    \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.Measure.restrict MeasureTheory.volume [[a, b]],\n      \u2016(\u230ax\u230b : \u2102)\u2016 \u2264 max \u2016a\u2016 \u2016b\u2016 + 1 := by\n  apply (MeasureTheory.ae_restrict_iff' measurableSet_Icc).mpr\n  refine MeasureTheory.ae_of_all _ (fun x hx \u21a6 ?_)\n  simp only [inf_le_iff, le_sup_iff, mem_Icc] at hx\n  simp only [norm_int, Real.norm_eq_abs]\n  have : |x| \u2264 max |a| |b| := by\n    cases' hx.1 with x_ge_a x_ge_b <;> cases' hx.2 with x_le_a x_le_b\n    \u00b7 rw [(by linarith : x = a)]; apply le_max_left\n    \u00b7 apply abs_le_max_abs_abs x_ge_a x_le_b\n    \u00b7 rw [max_comm]; apply abs_le_max_abs_abs x_ge_b x_le_a\n    \u00b7 rw [(by linarith : x = b)]; apply le_max_right\n  cases' abs_cases x with hx hx\n  \u00b7 rw [_root_.abs_of_nonneg <| by exact_mod_cast Int.floor_nonneg.mpr hx.2]\n    apply le_trans (Int.floor_le x) <| le_trans (hx.1 \u25b8 this) (by simp)\n  \u00b7 rw [_root_.abs_of_nonpos <| by exact_mod_cast Int.floor_nonpos hx.2.le]\n    linarith [(Int.lt_floor_add_one x).le]\n\nlemma integrability_aux\u2081 {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (\u230ax\u230b : \u2102)) MeasureTheory.volume a b := by\n  rw [intervalIntegrable_iff']\n  apply MeasureTheory.Measure.integrableOn_of_bounded ?_ ?_ integrability_aux\u2080\n  \u00b7 simp only [Real.volume_interval, ne_eq, ENNReal.ofReal_ne_top, not_false_eq_true]\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply Measurable.comp (by exact fun \u2983t\u2984 _ \u21a6 trivial) Int.measurable_floor\n\nlemma integrability_aux\u2082 {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (1 : \u2102) / 2 - x) MeasureTheory.volume a b :=\n  ContinuousOn.intervalIntegrable <| Continuous.continuousOn (by continuity)\n\nlemma integrability_aux {a b : \u211d} :\n    IntervalIntegrable (fun (x : \u211d) \u21a6 (\u230ax\u230b : \u2102) + 1 / 2 - x) MeasureTheory.volume a b := by\n  convert integrability_aux\u2081.add integrability_aux\u2082 using 2; ring\n\nlemma uIcc_subsets {a b c : \u211d} (hc : c \u2208 Icc a b) :\n    [[a, c]] \u2286 [[a, b]] \u2227 [[c, b]] \u2286 [[a, b]] := by\n  constructor <;> rw [uIcc_of_le ?_, uIcc_of_le ?_]\n  any_goals apply Icc_subset_Icc\n  all_goals linarith [hc.1, hc.2]\n\nlemma sum_eq_int_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (a_lt_b : a < b)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n      (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n        - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  let P := fun a\u2081 b\u2081 \u21a6 (\u2200 x \u2208 [[a\u2081, b\u2081]], HasDerivAt \u03c6 (deriv \u03c6 x) x) \u2192\n    (ContinuousOn (deriv \u03c6) [[a\u2081, b\u2081]]) \u2192\n    \u2211 n in Finset.Ioc \u230aa\u2081\u230b \u230ab\u2081\u230b, \u03c6 n =\n    (\u222b x in a\u2081..b\u2081, \u03c6 x) + (\u230ab\u2081\u230b + 1 / 2 - b\u2081) * \u03c6 b\u2081 - (\u230aa\u2081\u230b + 1 / 2 - a\u2081) * \u03c6 a\u2081\n      - \u222b x in a\u2081..b\u2081, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x\n  apply interval_induction P ?base ?step a b a_lt_b \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact fun _ _ _ k\u2081_le_a\u2081 a\u2081_lt_b\u2081 b\u2081_le_k\u2081 \u03c6Diff\u2081 deriv\u03c6Cont\u2081 \u21a6\n      sum_eq_int_deriv_aux \u27e8k\u2081_le_a\u2081, a\u2081_lt_b\u2081\u27e9 b\u2081_le_k\u2081 \u03c6Diff\u2081 deriv\u03c6Cont\u2081\n  \u00b7 intro a\u2081 k\u2081 b\u2081 a\u2081_lt_k\u2081 k\u2081_lt_b\u2081 ih\u2081 ih\u2082 \u03c6Diff\u2081 deriv\u03c6Cont\u2081\n    have subs := uIcc_subsets \u27e8a\u2081_lt_k\u2081.le, k\u2081_lt_b\u2081.le\u27e9\n    have s\u2081 := ih\u2081 (fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.1 hx) <| deriv\u03c6Cont\u2081.mono subs.1\n    have s\u2082 := ih\u2082 (fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.2 hx) <| deriv\u03c6Cont\u2081.mono subs.2\n    convert Mathlib.Tactic.LinearCombination.add_pf s\u2081 s\u2082 using 1\n    \u00b7 rw [\u2190 Finset.sum_Ioc_add_sum_Ioc]\n      simp only [Finset.mem_Icc, Int.floor_intCast, Int.le_floor]\n      exact \u27e8Int.cast_le.mp <| le_trans (Int.floor_le a\u2081) a\u2081_lt_k\u2081.le, k\u2081_lt_b\u2081.le\u27e9\n    \u00b7 set I\u2081 := \u222b (x : \u211d) in a\u2081..b\u2081, \u03c6 x\n      set I\u2082 := \u222b (x : \u211d) in a\u2081..k\u2081, \u03c6 x\n      set I\u2083 := \u222b (x : \u211d) in k\u2081..b\u2081, \u03c6 x\n      set J\u2081 := \u222b (x : \u211d) in a\u2081..b\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      set J\u2082 := \u222b (x : \u211d) in a\u2081..k\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      set J\u2083 := \u222b (x : \u211d) in k\u2081..b\u2081, (\u2191\u230ax\u230b + 1 / 2 - \u2191x) * deriv \u03c6 x\n      have hI : I\u2082 + I\u2083 = I\u2081 := by\n        apply intervalIntegral.integral_add_adjacent_intervals <;>\n        apply ContinuousOn.intervalIntegrable\n        \u00b7 exact HasDerivAt.continuousOn <| fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.1 hx\n        \u00b7 exact HasDerivAt.continuousOn <| fun x hx \u21a6 \u03c6Diff\u2081 x <| subs.2 hx\n      have hJ : J\u2082 + J\u2083 = J\u2081 := by\n        apply intervalIntegral.integral_add_adjacent_intervals <;>\n        apply IntervalIntegrable.mul_continuousOn\n        any_goals apply integrability_aux\n        \u00b7 exact deriv\u03c6Cont\u2081.mono subs.1\n        \u00b7 exact deriv\u03c6Cont\u2081.mono subs.2\n      rw [\u2190 hI, \u2190 hJ]; ring\n/-%%\n\\begin{proof}\\uses{sum_eq_int_deriv_aux}\\leanok\n  Apply Lemma \\ref{sum_eq_int_deriv_aux} in blocks of length $\\le 1$.\n\\end{proof}\n%%-/\n\nlemma xpos_of_uIcc {a b : \u2115} (ha : a \u2208 Ioo 0 b) {x : \u211d} (x_in : x \u2208 [[(a : \u211d), b]]) :\n    0 < x := by\n  rw [uIcc_of_le (by exact_mod_cast ha.2.le), mem_Icc] at x_in\n  linarith [(by exact_mod_cast ha.1 : (0 : \u211d) < a)]\n\nlemma neg_s_ne_neg_one {s : \u2102} (s_ne_one : s \u2260 1) : -s \u2260 -1 := fun hs \u21a6 s_ne_one <| neg_inj.mp hs\n\nlemma ZetaSum_aux1\u2081 {a b : \u2115} {s : \u2102} (s_ne_one : s \u2260 1) (ha : a \u2208 Ioo 0 b) :\n    (\u222b (x : \u211d) in a..b, 1 / (x : \u2102) ^ s) =\n    (b ^ (1 - s) - a ^ (1 - s)) / (1 - s) := by\n  convert integral_cpow (a := a) (b := b) (r := -s) ?_ using 1\n  \u00b7 refine intervalIntegral.integral_congr fun x hx \u21a6 one_div_cpow_eq ?_\n    exact (xpos_of_uIcc ha hx).ne'\n  \u00b7 norm_cast; rw [(by ring : -s + 1 = 1 - s)]\n  \u00b7 right; refine \u27e8neg_s_ne_neg_one s_ne_one, ?_\u27e9\n    exact fun hx \u21a6 (lt_self_iff_false 0).mp <| xpos_of_uIcc ha hx\n\nlemma ZetaSum_aux1\u03c6Diff {s : \u2102} {x : \u211d} (xpos : 0 < x) :\n    HasDerivAt (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) (deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) x) x := by\n  apply hasDerivAt_deriv_iff.mpr <| DifferentiableAt.div (differentiableAt_const _) ?_ ?_\n  \u00b7 exact Real.differentiableAt_cpow_const_of_ne s xpos\n  \u00b7 simp [cpow_eq_zero_iff, xpos.ne']\n\nlemma ZetaSum_aux1\u03c6deriv {s : \u2102} (s_ne_zero : s \u2260 0) {x : \u211d} (xpos : 0 < x) :\n    deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s) x = (fun (x : \u211d) \u21a6 -s * (x : \u2102) ^ (-(s + 1))) x := by\n  let r := -s - 1\n  have r_add1_ne_zero : r + 1 \u2260 0 := fun hr \u21a6 by simp [neg_ne_zero.mpr s_ne_zero, r] at hr\n  have r_ne_neg1 : r \u2260 -1 := fun hr \u21a6 (hr \u25b8 r_add1_ne_zero) <| by norm_num\n  have hasDeriv := hasDerivAt_ofReal_cpow xpos.ne' r_ne_neg1\n  have := hasDeriv.deriv \u25b8 deriv_const_mul (-s) (hasDeriv).differentiableAt\n  convert this using 2\n  \u00b7 ext y\n    by_cases y_zero : (y : \u2102) = 0\n    \u00b7 simp only [y_zero, ofReal_zero, ne_eq, s_ne_zero, not_false_eq_true, zero_cpow, div_zero,\n      r_add1_ne_zero, zero_div, mul_zero]\n    \u00b7 have : (y : \u2102) ^ s \u2260 0 := fun hy \u21a6 y_zero ((cpow_eq_zero_iff _ _).mp hy).1\n      field_simp [r, mul_assoc, \u2190 Complex.cpow_add]\n  \u00b7 ring_nf\n\nlemma ZetaSum_aux1deriv\u03c6Cont {s : \u2102} (s_ne_zero : s \u2260 0) {a b : \u2115} (ha : a \u2208 Ioo 0 b) :\n    ContinuousOn (deriv (fun (t : \u211d) \u21a6 1 / (t : \u2102) ^ s)) [[a, b]] := by\n  have : EqOn _ (fun (t : \u211d) \u21a6 -s * (t : \u2102) ^ (-(s + 1))) [[a, b]] :=\n    fun x hx \u21a6 ZetaSum_aux1\u03c6deriv s_ne_zero <| xpos_of_uIcc ha hx\n  refine ContinuousOn.congr ?_ this\n  refine (ContinuousOn.cpow_const continuous_ofReal.continuousOn ?_).const_smul (c := -s)\n  exact fun x hx \u21a6 ofReal_mem_slitPlane.mpr <| xpos_of_uIcc ha hx\n\n/-%%\n\\begin{lemma}[ZetaSum_aux1]\\label{ZetaSum_aux1}\\lean{ZetaSum_aux1}\\leanok\n  Let $0 < a < b$ be natural numbers and $s\\in \\C$ with $s \\ne 1$ and $s \\ne 0$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\frac{1}{n^s} =  \\frac{b^{1-s} - a^{1-s}}{1-s} + \\frac{b^{-s}-a^{-s}}{2} + s \\int_a^b \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma ZetaSum_aux1 {a b : \u2115} {s : \u2102} (s_ne_one : s \u2260 1) (s_ne_zero : s \u2260 0) (ha : a \u2208 Ioo 0 b) :\n    \u2211 n in Finset.Ioc (a : \u2124) b, 1 / (n : \u2102) ^ s =\n    (b ^ (1 - s) - a ^ (1 - s)) / (1 - s) + 1 / 2 * (1 / b ^ (s)) - 1 / 2 * (1 / a ^ s)\n      + s * \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1)) := by\n  let \u03c6 := fun (x : \u211d) \u21a6 1 / (x : \u2102) ^ s\n  let \u03c6' := fun (x : \u211d) \u21a6 -s * (x : \u2102) ^ (-(s + 1))\n  have xpos : \u2200 x \u2208 [[(a : \u211d), b]], 0 < x := fun x hx \u21a6 xpos_of_uIcc ha hx\n  have \u03c6Diff : \u2200 x \u2208 [[(a : \u211d), b]], HasDerivAt \u03c6 (deriv \u03c6 x) x := fun x hx \u21a6 ZetaSum_aux1\u03c6Diff (xpos x hx)\n  have \u03c6deriv : \u2200 x \u2208 [[(a : \u211d), b]], deriv \u03c6 x = \u03c6' x := by\n    exact fun x hx \u21a6 ZetaSum_aux1\u03c6deriv s_ne_zero (xpos x hx)\n  have deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]] := ZetaSum_aux1deriv\u03c6Cont s_ne_zero ha\n  convert sum_eq_int_deriv (by exact_mod_cast ha.2) \u03c6Diff deriv\u03c6Cont using 1\n  \u00b7 congr <;> simp only [Int.floor_natCast]\n  \u00b7 rw [Int.floor_natCast, Int.floor_natCast, \u2190 intervalIntegral.integral_const_mul]\n    simp_rw [mul_div, \u2190 mul_div, ZetaSum_aux1\u2081 s_ne_one ha]\n    conv => rhs; rw [sub_eq_add_neg]\n    congr; any_goals norm_cast; simp only [one_div, add_sub_cancel_left]\n    rw [\u2190 intervalIntegral.integral_neg, intervalIntegral.integral_congr]\n    intro x hx; simp_rw [\u03c6deriv x hx, \u03c6']; ring_nf\n/-%%\n\\begin{proof}\\uses{sum_eq_int_deriv}\\leanok\n  Apply Lemma \\ref{sum_eq_int_deriv} to the function $x \\mapsto x^{-s}$.\n\\end{proof}\n%%-/\n\nlemma ZetaSum_aux1_1' {a b x : \u211d} (apos : 0 < a) (hx : x \u2208 Icc a b)\n    : 0 < x := lt_of_lt_of_le apos hx.1\n\nlemma ZetaSum_aux1_1 {a b x : \u211d} (apos : 0 < a) (a_lt_b : a < b) (hx : x \u2208 [[a,b]])\n    : 0 < x :=  lt_of_lt_of_le apos (uIcc_of_le a_lt_b.le \u25b8 hx).1\n\nlemma ZetaSum_aux1_2 {a b : \u211d} {c : \u211d} (apos : 0 < a) (a_lt_b : a < b)\n    (h : c \u2260 0 \u2227 0 \u2209 [[a, b]]) :\n    \u222b (x : \u211d) in a..b, 1 / x ^ (c+1) = (a ^ (-c) - b ^ (-c)) / c := by\n  rw [(by ring : (a ^ (-c) - b ^ (-c)) / c = (b ^ (-c) - a ^ (-c)) / (-c))]\n  have := integral_rpow (a := a) (b := b) (r := -c-1) (Or.inr \u27e8by simp [h.1], h.2\u27e9)\n  simp only [sub_add_cancel] at this\n  rw [\u2190 this]\n  apply intervalIntegral.integral_congr\n  intro x hx\n  have : 0 \u2264 x := (ZetaSum_aux1_1 apos a_lt_b hx).le\n  simp [div_rpow_eq_rpow_neg _ _ _ this, sub_eq_add_neg, add_comm]\n\nlemma ZetaSum_aux1_3a (x : \u211d) : -(1/2) < \u230a x \u230b + 1/2 - x := by\n  norm_num [\u2190 add_assoc]; linarith [sub_pos_of_lt (Int.lt_floor_add_one x)]\n\nlemma ZetaSum_aux1_3b (x : \u211d) : \u230ax\u230b + 1/2 - x \u2264 1/2 := by\n  ring_nf; exact add_le_of_nonpos_right <| sub_nonpos.mpr (Int.floor_le x)\n\nlemma ZetaSum_aux1_3 (x : \u211d) : |(\u230ax\u230b + 1/2 - x)| \u2264 1/2 :=\n  abs_le.mpr \u27e8le_of_lt (ZetaSum_aux1_3a x), ZetaSum_aux1_3b x\u27e9\n\nlemma ZetaSum_aux1_4' (x : \u211d) (hx : 0 < x) (s : \u2102) :\n      \u2016(\u230ax\u230b + 1 / 2 - (x : \u211d)) / (x : \u2102) ^ (s + 1)\u2016 =\n      |\u230ax\u230b + 1 / 2 - x| / x ^ ((s + 1).re) := by\n  simp [map_div\u2080, abs_ofReal, Complex.abs_cpow_eq_rpow_re_of_pos hx, \u2190 abs_ofReal]\n\nlemma ZetaSum_aux1_4 {a b : \u211d} (apos : 0 < a) (a_lt_b : a < b) {s : \u2102} :\n  \u222b (x : \u211d) in a..b, \u2016(\u2191\u230ax\u230b + (1 : \u211d) / 2 - \u2191x) / (x : \u2102) ^ (s + 1)\u2016 =\n    \u222b (x : \u211d) in a..b, |\u230ax\u230b + 1 / 2 - x| / x ^ (s + 1).re := by\n  apply intervalIntegral.integral_congr\n  exact fun x hx \u21a6 ZetaSum_aux1_4' x (ZetaSum_aux1_1 apos a_lt_b hx) s\n\n", "theoremStatement": "lemma ZetaSum_aux1_5a {a b : \u211d} (apos : 0 < a) {s : \u2102} (x : \u211d)\n  (h : x \u2208 Icc a b) : |\u2191\u230ax\u230b + 1 / 2 - x| / x ^ (s.re + 1) \u2264 1 / x ^ (s.re + 1) ", "theoremName": "ZetaSum_aux1_5a", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": 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"Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply div_le_div_of_nonneg_right _ _\n  \u00b7 exact le_trans (ZetaSum_aux1_3 x) (by norm_num)\n  \u00b7 apply Real.rpow_nonneg <| le_of_lt (ZetaSum_aux1_1' apos h)", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 160}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module aux_results\n-/\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Integrals\nimport Mathlib.Analysis.SpecialFunctions.Log.Basic\nimport Mathlib.Analysis.SpecialFunctions.NonIntegrable\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal\n\nnoncomputable section\n\nopen scoped BigOperators ArithmeticFunction\n\nopen Nat ArithmeticFunction Finset\n\n\nnamespace ArithmeticFunction.IsMultiplicative\n\nvariable {R : Type*}\n\ntheorem mult_lcm_eq_of_ne_zero [CommGroupWithZero R] (f : ArithmeticFunction R) (h_mult : f.IsMultiplicative) (x y : \u2115)\n    (hf : f (x.gcd y) \u2260 0) :\n    f (x.lcm y) = f x * f y / f (x.gcd y) := by\n  rw [\u2190h_mult.lcm_apply_mul_gcd_apply]\n  field_simp\n\ntheorem prod_factors_of_mult (f : ArithmeticFunction \u211d) (h_mult : ArithmeticFunction.IsMultiplicative f) {l : \u2115} (hl : Squarefree l) :\n    \u220f a : \u2115 in l.primeFactors, f a = f l := by\n  rw [\u2190IsMultiplicative.map_prod_of_subset_primeFactors h_mult l _ Finset.Subset.rfl,\n    Nat.prod_primeFactors_of_squarefree hl]\n\nend ArithmeticFunction.IsMultiplicative\n\nnamespace Aux\ntheorem sum_over_dvd_ite {\u03b1 : Type _} [Ring \u03b1] {P : \u2115} (hP : P \u2260 0) {n : \u2115} (hn : n \u2223 P)\n    {f : \u2115 \u2192 \u03b1} : \u2211 d in n.divisors, f d = \u2211 d in P.divisors, if d \u2223 n then f d else 0 :=\n  by\n  rw [\u2190Finset.sum_filter, Nat.divisors_filter_dvd_of_dvd hP hn]\n\ntheorem sum_intro {\u03b1 M: Type _} [AddCommMonoid M] [DecidableEq \u03b1] (s : Finset \u03b1) {f : \u03b1 \u2192 M} (d : \u03b1)\n     (hd : d \u2208 s) :\n    f d = \u2211 k in s, if k = d then f k else 0 := by\n  trans (\u2211 k in s, if k = d then f d else 0)\n  \u00b7 rw [sum_eq_single_of_mem d hd]\n    rw [if_pos rfl]\n    intro _ _ h; rw [if_neg h]\n  apply sum_congr rfl; intro k _; apply if_ctx_congr Iff.rfl _ (fun _ => rfl)\n  intro h; rw [h]\n\ntheorem ite_sum_zero {p : Prop} [Decidable p] (s : Finset \u2115) (f : \u2115 \u2192 \u211d) :\n    (if p then (\u2211 x in s, f x) else 0) = \u2211 x in s, if p then f x else 0 := by\n  split_ifs <;> simp\n\ntheorem conv_lambda_sq_larger_sum (f : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u211d) (n : \u2115) :\n    (\u2211 d in n.divisors,\n        \u2211 d1 in d.divisors,\n          \u2211 d2 in d.divisors, if d = Nat.lcm d1 d2 then f d1 d2 d else 0) =\n      \u2211 d in n.divisors,\n        \u2211 d1 in n.divisors,\n          \u2211 d2 in n.divisors, if d = Nat.lcm d1 d2 then f d1 d2 d else 0 := by\n  apply sum_congr rfl; intro d hd\n  rw [mem_divisors] at hd\n  simp_rw [\u2190Nat.divisors_filter_dvd_of_dvd hd.2 hd.1, sum_filter, \u2190ite_and, ite_sum_zero, \u2190ite_and]\n  congr with d1\n  congr with d2\n  congr\n  rw [eq_iff_iff]\n  refine \u27e8fun \u27e8_, _, h\u27e9 \u21a6 h, ?_\u27e9\n  rintro rfl\n  exact \u27e8Nat.dvd_lcm_left d1 d2, Nat.dvd_lcm_right d1 d2, rfl\u27e9\n\ntheorem moebius_inv_dvd_lower_bound (l m : \u2115) (hm : Squarefree m) :\n    (\u2211 d in m.divisors, if l \u2223 d then (\u03bc d:\u2124) else 0) = if l = m then (\u03bc l:\u2124) else 0 := by\n  have hm_pos : 0 < m := Nat.pos_of_ne_zero $ Squarefree.ne_zero hm\n  revert hm\n  revert m\n  apply (ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on {n | Squarefree n} (fun _ _ => Squarefree.squarefree_of_dvd)).mpr\n  intro m hm_pos hm\n  rw [sum_divisorsAntidiagonal' (f:= fun x y => \u03bc x \u2022 if l=y then \u03bc l else 0)]--\n  by_cases hl : l \u2223 m\n  \u00b7 rw [if_pos hl, sum_eq_single l]\n    \u00b7 have hmul : m / l * l = m := Nat.div_mul_cancel hl\n      rw [if_pos rfl, smul_eq_mul, \u2190isMultiplicative_moebius.map_mul_of_coprime,\n        hmul]\n\n      apply coprime_of_squarefree_mul; rw [hmul]; exact hm\n    \u00b7 intro d _ hdl; rw[if_neg $ hdl.symm, smul_zero]\n    \u00b7 intro h; rw[mem_divisors] at h; exfalso; exact h \u27e8hl, (Nat.ne_of_lt hm_pos).symm\u27e9\n  \u00b7 rw [if_neg hl, sum_eq_zero]; intro d hd\n    rw [if_neg, smul_zero]\n    by_contra h; rw [\u2190h] at hd; exact hl (dvd_of_mem_divisors hd)\n\n\ntheorem moebius_inv_dvd_lower_bound' {P : \u2115} (hP : Squarefree P) (l m : \u2115) (hm : m \u2223 P) :\n    (\u2211 d in P.divisors, if l \u2223 d \u2227 d \u2223 m then \u03bc d else 0) = if l = m then \u03bc l else 0 := by\n  rw [\u2190moebius_inv_dvd_lower_bound _ _ (Squarefree.squarefree_of_dvd hm hP),\n    sum_over_dvd_ite hP.ne_zero hm]\n  simp_rw[ite_and, \u2190sum_filter, filter_comm]\n\ntheorem moebius_inv_dvd_lower_bound_real {P : \u2115} (hP : Squarefree P) (l m : \u2115) (hm : m \u2223 P) :\n    (\u2211 d in P.divisors, if l \u2223 d \u2227 d \u2223 m then (\u03bc d : \u211d) else 0) = if l = m then (\u03bc l : \u211d) else 0 := by\n  norm_cast\n  apply moebius_inv_dvd_lower_bound' hP l m hm\n\ntheorem gcd_dvd_mul (m n : \u2115) : m.gcd n \u2223 m * n := by\n  calc\n    m.gcd n \u2223 m := Nat.gcd_dvd_left m n\n    _ \u2223 m * n := \u27e8n, rfl\u27e9\n\ntheorem multiplicative_zero_of_zero_dvd (f : ArithmeticFunction \u211d) (h_mult : IsMultiplicative f) {m n : \u2115}\n    (h_sq : Squarefree n) (hmn : m \u2223 n) (h_zero : f m = 0) : f n = 0 := by\n  rcases hmn with \u27e8k, rfl\u27e9\n  simp only [MulZeroClass.zero_mul, eq_self_iff_true, h_mult.map_mul_of_coprime\n    (coprime_of_squarefree_mul h_sq), h_zero]\n\ntheorem primeDivisors_nonempty (n : \u2115) (hn : 2 \u2264 n) : n.primeFactors.Nonempty := by\n  unfold Finset.Nonempty\n  simp_rw[Nat.mem_primeFactors_of_ne_zero (by positivity)]\n  apply Nat.exists_prime_and_dvd (by linarith)\n\ntheorem div_mult_of_dvd_squarefree (f : ArithmeticFunction \u211d) (h_mult : IsMultiplicative f) (l d : \u2115) (hdl : d \u2223 l)\n    (hl : Squarefree l) (hd : f d \u2260 0) : f l / f d = f (l / d) := by\n  apply div_eq_of_eq_mul hd\n  rw [\u2190 h_mult.right, Nat.div_mul_cancel hdl]\n  apply coprime_of_squarefree_mul\n  convert hl\n  exact Nat.div_mul_cancel hdl\n\ntheorem inv_sub_antitoneOn_gt {R : Type*} [LinearOrderedField R] (c : R) :\n    AntitoneOn (fun x:R \u21a6 (x-c)\u207b\u00b9) (Set.Ioi c) := by\n  refine antitoneOn_iff_forall_lt.mpr ?_\n  intro a ha b hb hab\n  rw [Set.mem_Ioi] at ha hb\n  gcongr; linarith\n\ntheorem inv_sub_antitoneOn_Icc {R : Type*} [LinearOrderedField R]  (a b c: R) (ha : c < a) :\n    AntitoneOn (fun x \u21a6 (x-c)\u207b\u00b9) (Set.Icc a b) := by\n  by_cases hab : a \u2264 b\n  \u00b7 exact inv_sub_antitoneOn_gt c |>.mono <| (Set.Icc_subset_Ioi_iff hab).mpr ha\n  \u00b7 simp [hab, Set.Subsingleton.antitoneOn]\n\ntheorem inv_antitoneOn_pos {R : Type*} [LinearOrderedField R] :\n    AntitoneOn (fun x:R \u21a6 x\u207b\u00b9) (Set.Ioi 0) := by\n  convert inv_sub_antitoneOn_gt (R:=R) 0; ring\n\ntheorem inv_antitoneOn_Icc {R : Type*} [LinearOrderedField R] (a b : R) (ha : 0 < a) :\n    AntitoneOn (fun x \u21a6 x\u207b\u00b9) (Set.Icc a b) := by\n  convert inv_sub_antitoneOn_Icc a b 0 ha; ring\n\n", "theoremStatement": "theorem log_add_one_le_sum_inv (n : \u2115) :\n    Real.log \u2191(n+1) \u2264 \u2211 d in Finset.Icc 1 n, (d:\u211d)\u207b\u00b9 ", "theoremName": "Aux.log_add_one_le_sum_inv", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/AuxResults.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults.jsonl", "positionMetadata": {"lineInFile": 158, "tokenPositionInFile": 6462, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, 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"Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  calc _ = \u222b x in (1)..\u2191(n+1), x\u207b\u00b9 := ?_\n       _ = \u222b x in (1:\u2115)..\u2191(n+1), x\u207b\u00b9 := ?_\n       _ \u2264 _ := ?_\n  \u00b7 rw[integral_inv (by simp[(show \u00ac (1:\u211d) \u2264 0 by norm_num)] )]; congr; ring\n  \u00b7 congr; norm_num\n  \u00b7 apply AntitoneOn.integral_le_sum_Ico (by norm_num)\n    apply inv_antitoneOn_Icc\n    norm_num", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 302}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\n", "theoremStatement": "lemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s ", "theoremName": "div_rpow_neg_eq_rpow_div", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "56010d7e2c5806411c299f1e0e47d68f4a3edd60", "date": "2024-04-25"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": "PrimeNumberTheoremAnd.ZetaBounds.jsonl", "positionMetadata": {"lineInFile": 29, "tokenPositionInFile": 1061, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 56, "importedModules": ["Init.Prelude", "Init.Coe", 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"Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", 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"Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 96}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]\n\nlemma Filter.BigO_zero_atTop_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[atTop] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioi_mem_atTop b] with c hc; replace hc := mem_Ioi.mp hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h]\n\n-- steal coerction lemmas from EulerProducts.Auxiliary because of build issues, and add new ones\nnamespace Complex\n-- see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Differentiability.20of.20the.20natural.20map.20.E2.84.9D.20.E2.86.92.20.E2.84.82/near/418095234\n\nlemma hasDerivAt_ofReal (x : \u211d) : HasDerivAt ofReal' 1 x :=\n  HasDerivAt.ofReal_comp <| hasDerivAt_id x\n\nlemma deriv_ofReal (x : \u211d) : deriv ofReal' x = 1 :=\n  (hasDerivAt_ofReal x).deriv\n\nlemma differentiableAt_ofReal (x : \u211d) : DifferentiableAt \u211d ofReal' x :=\n  (hasDerivAt_ofReal x).differentiableAt\n\nlemma differentiable_ofReal : Differentiable \u211d ofReal' :=\n  ofRealCLM.differentiable\n\nend Complex\n\nlemma DifferentiableAt.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    DifferentiableAt \u211d (fun x : \u211d \u21a6 e x) z :=\n  hf.hasDerivAt.comp_ofReal.differentiableAt\n\nlemma Differentiable.comp_ofReal {e : \u2102 \u2192 \u2102} (h : Differentiable \u2102 e) :\n    Differentiable \u211d (fun x : \u211d \u21a6 e x) :=\n  fun _ \u21a6 h.differentiableAt.comp_ofReal\n\nlemma DifferentiableAt.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} (hf : DifferentiableAt \u211d f z) :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z :=\n  hf.hasDerivAt.ofReal_comp.differentiableAt\n\nlemma Differentiable.ofReal_comp {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) :=\n  fun _ \u21a6 hf.differentiableAt.ofReal_comp\n\nopen Complex ContinuousLinearMap in\nlemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} {u : \u2102}\n    (hf : HasDerivAt (fun y \u21a6 (f y : \u2102)) u z) :\n    \u2203 u' : \u211d, u = u' \u2227 HasDerivAt f u' z := by\n  lift u to \u211d\n  \u00b7 have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv\n    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H\n    rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H\n  refine \u27e8u, rfl, ?_\u27e9\n  convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt\n  rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]\n\nlemma DifferentiableAt.ofReal_comp_iff {z : \u211d} {f : \u211d \u2192 \u211d} :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z \u2194 DifferentiableAt \u211d f z := by\n  refine \u27e8fun H \u21a6 ?_, ofReal_comp\u27e9\n  obtain \u27e8u, _, hu\u2082\u27e9 := H.hasDerivAt.of_hasDerivAt_ofReal_comp\n  exact HasDerivAt.differentiableAt hu\u2082\n\nlemma Differentiable.ofReal_comp_iff {f : \u211d \u2192 \u211d} :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) \u2194 Differentiable \u211d f :=\n  forall_congr' fun _ \u21a6 DifferentiableAt.ofReal_comp_iff\n\nlemma deriv.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} :\n    deriv (fun (y : \u211d) \u21a6 (f y : \u2102)) z = deriv f z := by\n  by_cases hf : DifferentiableAt \u211d f z\n  \u00b7 exact hf.hasDerivAt.ofReal_comp.deriv\n  \u00b7 have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf\n    rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt <| hf',\n      Complex.ofReal_zero]\n\n", "theoremStatement": "lemma deriv.ofReal_comp' {f : \u211d \u2192 \u211d} :\n    deriv (fun x : \u211d \u21a6 (f x : \u2102)) = (fun x \u21a6 ((deriv f) x : \u2102)) ", "theoremName": "deriv.ofReal_comp'", "fileCreated": {"commit": 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"Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", 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"Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  funext fun _ \u21a6 deriv.ofReal_comp", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 37}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\n", "theoremStatement": "lemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) ", "theoremName": "bound_sum_log", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "86134aa14af3a4d4f6533259233b10396c33be51", "date": "2024-03-24"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1351, "tokenPositionInFile": 66324, "theoremPositionInFile": 120}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 12, "repositoryPremises": true, 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"Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self", "proofType": "tactic", "proofLengthLines": 66, "proofLengthTokens": 2853}}
+{"srcContext": "/-\nCopyright (c) 2024 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.Calculus.LineDeriv.Basic\nimport Mathlib.MeasureTheory.Integral.IntegralEqImproper\n-- import Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\n\n/-!\n# Integration by parts for line derivatives\n\nLet `f, g : E \u2192 \u211d` be two differentiable functions on a real vector space endowed with a Haar\nmeasure. Then `\u222b f * g' = - \u222b f' * g`, where `f'` and `g'` denote the derivatives of `f` and `g`\nin a given direction `v`, provided that `f * g`, `f' * g` and `f * g'` are all integrable.\n\nIn this file, we prove this theorem as well as more general versions where the multiplication is\nreplaced by a general continuous bilinear form, giving versions both for the line derivative and\nthe Fr\u00e9chet derivative. These results are derived from the one-dimensional version and a Fubini\nargument.\n\n## Main statements\n\n* `integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable`: integration by parts\n  in terms of line derivatives, with `HasLineDerivAt` assumptions and general bilinear form.\n* `integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable`: integration by parts\n  in terms of Fr\u00e9chet derivatives, with `HasFDerivAt` assumptions and general bilinear form.\n* `integral_bilinear_fderiv_right_eq_neg_left_of_integrable`: integration by parts\n  in terms of Fr\u00e9chet derivatives, written with `fderiv` assumptions and general bilinear form.\n* `integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable`: integration by parts for scalar\n  action, in terms of Fr\u00e9chet derivatives, written with `fderiv` assumptions.\n* `integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable`: integration by parts for scalar\n  multiplication, in terms of Fr\u00e9chet derivatives, written with `fderiv` assumptions.\n\n-/\n\nopen MeasureTheory Measure FiniteDimensional\n\nvariable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [NormedAddCommGroup F]\n  [NormedSpace \u211d F] [NormedAddCommGroup G] [NormedSpace \u211d G] [NormedAddCommGroup W]\n  [NormedSpace \u211d W] [MeasurableSpace E] [BorelSpace E] {\u03bc : Measure E}\n\nlemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 [SigmaFinite \u03bc]\n    {f f' : E \u00d7 \u211d \u2192 F} {g g' : E \u00d7 \u211d \u2192 G} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) (\u03bc.prod volume))\n    (hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) (\u03bc.prod volume))\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) (\u03bc.prod volume))\n    (hf : \u2200 x, HasLineDerivAt \u211d f (f' x) x (0, 1)) (hg : \u2200 x, HasLineDerivAt \u211d g (g' x) x (0, 1)) :\n    \u222b x, B (f x) (g' x) \u2202(\u03bc.prod volume) = - \u222b x, B (f' x) (g x) \u2202(\u03bc.prod volume) := calc\n  \u222b x, B (f x) (g' x) \u2202(\u03bc.prod volume)\n    = \u222b x, (\u222b t, B (f (x, t)) (g' (x, t))) \u2202\u03bc := integral_prod _ hfg'\n  _ = \u222b x, (- \u222b t, B (f' (x, t)) (g (x, t))) \u2202\u03bc := by\n    apply integral_congr_ae\n    filter_upwards [hf'g.prod_right_ae, hfg'.prod_right_ae, hfg.prod_right_ae]\n      with x hf'gx hfg'x  hfgx\n    apply integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable ?_ ?_ hfg'x hf'gx hfgx\n    \u00b7 intro t\n      convert (hf (x, t)).scomp_of_eq t ((hasDerivAt_id t).add (hasDerivAt_const t (-t))) (by simp)\n        <;> simp\n    \u00b7 intro t\n      convert (hg (x, t)).scomp_of_eq t ((hasDerivAt_id t).add (hasDerivAt_const t (-t))) (by simp)\n        <;> simp\n  _ = - \u222b x, B (f' x) (g x) \u2202(\u03bc.prod volume) := by rw [integral_neg, integral_prod _ hf'g]\n\nlemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2\n    [FiniteDimensional \u211d E] {\u03bc : Measure (E \u00d7 \u211d)} [IsAddHaarMeasure \u03bc]\n    {f f' : E \u00d7 \u211d \u2192 F} {g g' : E \u00d7 \u211d \u2192 G} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) \u03bc)\n    (hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) \u03bc)\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) \u03bc)\n    (hf : \u2200 x, HasLineDerivAt \u211d f (f' x) x (0, 1)) (hg : \u2200 x, HasLineDerivAt \u211d g (g' x) x (0, 1)) :\n    \u222b x, B (f x) (g' x) \u2202\u03bc = - \u222b x, B (f' x) (g x) \u2202\u03bc := by\n  let \u03bd : Measure E := addHaar\n  have A : \u03bd.prod volume = (addHaarScalarFactor (\u03bd.prod volume) \u03bc) \u2022 \u03bc :=\n    isAddLeftInvariant_eq_smul _ _\n  have Hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) (\u03bd.prod volume) := by\n    rw [A]; exact hf'g.smul_measure_nnreal\n  have Hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) (\u03bd.prod volume) := by\n    rw [A]; exact hfg'.smul_measure_nnreal\n  have Hfg : Integrable (fun x \u21a6 B (f x) (g x)) (\u03bd.prod volume) := by\n    rw [A]; exact hfg.smul_measure_nnreal\n  rw [isAddLeftInvariant_eq_smul \u03bc (\u03bd.prod volume)]\n  simp [integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 Hf'g Hfg' Hfg hf hg]\n\nvariable [FiniteDimensional \u211d E] [IsAddHaarMeasure \u03bc]\n\n", "theoremStatement": "/-- **Integration by parts for line derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives\nof `f` and `g` in a given direction `v`, then `\u222b B f g' = - \u222b B f' g`. -/\ntheorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable\n    {f f' : E \u2192 F} {g g' : E \u2192 G} {v : E} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) \u03bc) (hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) \u03bc)\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) \u03bc)\n    (hf : \u2200 x, HasLineDerivAt \u211d f (f' x) x v) (hg : \u2200 x, HasLineDerivAt \u211d g (g' x) x v) :\n    \u222b x, B (f x) (g' x) \u2202\u03bc = - \u222b x, B (f' x) (g x) \u2202\u03bc ", "theoremName": "integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable", "fileCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "theoremCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts.jsonl", "positionMetadata": {"lineInFile": 87, "tokenPositionInFile": 4674, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, 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"Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  by_cases hW : CompleteSpace W; swap\n  \u00b7 simp [integral, hW]\n  rcases eq_or_ne v 0 with rfl|hv\n  \u00b7 have Hf' x : f' x = 0 := by\n      simpa [(hasLineDerivAt_zero (f := f) (x := x)).lineDeriv] using (hf x).lineDeriv.symm\n    have Hg' x : g' x = 0 := by\n      simpa [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv] using (hg x).lineDeriv.symm\n    simp [Hf', Hg']\n  have : Nontrivial E := nontrivial_iff.2 \u27e8v, 0, hv\u27e9\n  let n := finrank \u211d E\n  let E' := Fin (n - 1) \u2192 \u211d\n  obtain \u27e8L, hL\u27e9 : \u2203 L : E \u2243L[\u211d] (E' \u00d7 \u211d), L v = (0, 1) := by\n    have : finrank \u211d (E' \u00d7 \u211d) = n := by simpa [this, E'] using Nat.sub_add_cancel finrank_pos\n    have L\u2080 : E \u2243L[\u211d] (E' \u00d7 \u211d) := (ContinuousLinearEquiv.ofFinrankEq this).symm\n    obtain \u27e8M, hM\u27e9 : \u2203 M : (E' \u00d7 \u211d) \u2243L[\u211d] (E' \u00d7 \u211d), M (L\u2080 v) = (0, 1) := by\n      apply SeparatingDual.exists_continuousLinearEquiv_apply_eq\n      \u00b7 simpa using hv\n      \u00b7 simp\n    exact \u27e8L\u2080.trans M, by simp [hM]\u27e9\n  let \u03bd := Measure.map L \u03bc\n  suffices H : \u222b (x : E' \u00d7 \u211d), (B (f (L.symm x))) (g' (L.symm x)) \u2202\u03bd =\n      -\u222b (x : E' \u00d7 \u211d), (B (f' (L.symm x))) (g (L.symm x)) \u2202\u03bd by\n    have : \u03bc = Measure.map L.symm \u03bd := by\n      simp [Measure.map_map L.symm.continuous.measurable L.continuous.measurable]\n    have hL : ClosedEmbedding L.symm := L.symm.toHomeomorph.closedEmbedding\n    simpa [this, hL.integral_map] using H\n  have L_emb : MeasurableEmbedding L := L.toHomeomorph.measurableEmbedding\n  apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2\n  \u00b7 simpa [L_emb.integrable_map_iff, Function.comp] using hf'g\n  \u00b7 simpa [L_emb.integrable_map_iff, Function.comp] using hfg'\n  \u00b7 simpa [L_emb.integrable_map_iff, Function.comp] using hfg\n  \u00b7 intro x\n    have : f = (f \u2218 L.symm) \u2218 (L : E \u2192\u2097[\u211d] (E' \u00d7 \u211d)) := by ext y; simp\n    specialize hf (L.symm x)\n    rw [this] at hf\n    convert hf.of_comp using 1\n    \u00b7 simp\n    \u00b7 simp [\u2190 hL]\n  \u00b7 intro x\n    have : g = (g \u2218 L.symm) \u2218 (L : E \u2192\u2097[\u211d] (E' \u00d7 \u211d)) := by ext y; simp\n    specialize hg (L.symm x)\n    rw [this] at hg\n    convert hg.of_comp using 1\n    \u00b7 simp\n    \u00b7 simp [\u2190 hL]", "proofType": "tactic", "proofLengthLines": 45, "proofLengthTokens": 2052}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\n", "theoremStatement": "lemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) ", "theoremName": "limiting_fourier", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "9bbe1ab56acf8cd5ebda1e3863028775fb2508e6", "date": "2024-01-24"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 976, "tokenPositionInFile": 49167, "theoremPositionInFile": 82}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 10, "numPremises": 272, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", 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"Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 308}}
+{"srcContext": "/-\nCopyright (c) 2024 David Loeffler. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex Kontorovich, David Loeffler, Heather Macbeth\n-/\nimport Mathlib.Analysis.Calculus.ParametricIntegral\nimport Mathlib.Analysis.Fourier.AddCircle\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Calculus.FDeriv.Analytic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space\n\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\n\n/-!\n# Derivatives of the Fourier transform\n\nIn this file we compute the Fr\u00e9chet derivative of the Fourier transform of `f`, where `f` is a\nfunction such that both `f` and `v \u21a6 \u2016v\u2016 * \u2016f v\u2016` are integrable. Here the Fourier transform is\nunderstood as an operator `(V \u2192 E) \u2192 (W \u2192 E)`, where `V` and `W` are normed `\u211d`-vector spaces\nand the Fourier transform is taken with respect to a continuous `\u211d`-bilinear\npairing `L : V \u00d7 W \u2192 \u211d` and a given reference measure `\u03bc`.\n\nWe also investigate higher derivatives: Assuming that `\u2016v\u2016^n * \u2016f v\u2016` is integrable, we show\nthat the Fourier transform of `f` is `C^n`.\n\nWe also study in a parallel way the Fourier transform of the derivative, which is obtained by\ntensoring the Fourier transform of the original function with the bilinear form.\n\nWe give specialized versions of these results on inner product spaces (where `L` is the scalar\nproduct) and on the real line, where we express the one-dimensional derivative in more concrete\nterms, as the Fourier transform of `-2\u03c0I x * f x` (or `(-2\u03c0I x)^n * f x` for higher derivatives).\n\n## Main definitions and results\n\nWe introduce two convenience definitions:\n\n* `VectorFourier.fourierSMulRight L f`: given `f : V \u2192 E` and `L` a bilinear pairing\n  between `V` and `W`, then this is the function `fun v \u21a6 -(2 * \u03c0 * I) (L v \u2b1d) \u2022 f v`,\n  from `V` to `Hom (W, E)`.\n  This is essentially `ContinousLinearMap.smulRight`, up to the factor `- 2\u03c0I` designed to make sure\n  that the Fourier integral of `fourierSMulRight L f` is the derivative of the Fourier\n  integral of `f`.\n* `VectorFourier.fourierPowSMulRight` is the higher order analogue for higher derivatives:\n  `fourierPowSMulRight L f v n` is informally `(-(2 * \u03c0 * I))^n (L v \u2b1d)^n \u2022 f v`, in\n  the space of continuous multilinear maps `W [\u00d7n]\u2192L[\u211d] E`.\n\nWith these definitions, the statements read as follows, first in a general context\n(arbitrary `L` and `\u03bc`):\n\n* `VectorFourier.hasFDerivAt_fourierIntegral`: the Fourier integral of `f` is differentiable, with\n    derivative the Fourier integral of `fourierSMulRight L f`.\n* `VectorFourier.differentiable_fourierIntegral`: the Fourier integral of `f` is differentiable.\n* `VectorFourier.fderiv_fourierIntegral`: formula for the derivative of the Fourier integral of `f`.\n* `VectorFourier.fourierIntegral_fderiv`: formula for the Fourier integral of the derivative of `f`.\n* `VectorFourier.hasFTaylorSeriesUpTo_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` has an explicit Taylor series up to order `N`, given by the Fourier\n  integrals of `fun v \u21a6 fourierPowSMulRight L f v n`.\n* `VectorFourier.contDiff_fourierIntegral`: under suitable integrability conditions,\n  the Fourier integral of `f` is `C^n`.\n* `VectorFourier.iteratedFDeriv_fourierIntegral`: under suitable integrability conditions,\n  explicit formula for the `n`-th derivative of the Fourier integral of `f`, as the Fourier\n  integral of `fun v \u21a6 fourierPowSMulRight L f v n`.\n\nThese statements are then specialized to the case of the usual Fourier transform on\nfinite-dimensional inner product spaces with their canonical Lebesgue measure (covering in\nparticular the case of the real line), replacing the namespace `VectorFourier` by\nthe namespace `Real` in the above statements.\n\nWe also give specialized versions of the one-dimensional real derivative (and iterated derivative)\nin `Real.deriv_fourierIntegral` and `Real.iteratedDeriv_fourierIntegral`.\n-/\n\nnoncomputable section\n\nopen Real Complex MeasureTheory Filter TopologicalSpace\n\nopen scoped FourierTransform Topology BigOperators\n\n-- without this local instance, Lean tries first the instance\n-- `secondCountableTopologyEither_of_right` (whose priority is 100) and takes a very long time to\n-- fail. Since we only use the left instance in this file, we make sure it is tried first.\nattribute [local instance 101] secondCountableTopologyEither_of_left\n\nnamespace Real\n\nlemma differentiable_fourierChar : Differentiable \u211d (\ud835\udc1e \u00b7 : \u211d \u2192 \u2102) :=\n  fun x \u21a6 (Real.hasDerivAt_fourierChar x).differentiableAt\n\nlemma deriv_fourierChar (x : \u211d) : deriv (\ud835\udc1e \u00b7 : \u211d \u2192 \u2102) x = 2 * \u03c0 * I * \ud835\udc1e x :=\n  (Real.hasDerivAt_fourierChar x).deriv\n\nvariable {V W : Type*} [NormedAddCommGroup V] [NormedSpace \u211d V]\n  [NormedAddCommGroup W] [NormedSpace \u211d W] (L : V \u2192L[\u211d] W \u2192L[\u211d] \u211d)\n\nlemma hasFDerivAt_fourierChar_neg_bilinear_right (v : V) (w : W) :\n    HasFDerivAt (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102))\n      ((-2 * \u03c0 * I * \ud835\udc1e (-L v w)) \u2022 (ofRealCLM \u2218L (L v))) w := by\n  have ha : HasFDerivAt (fun w' : W \u21a6 L v w') (L v) w := ContinuousLinearMap.hasFDerivAt (L v)\n  convert (hasDerivAt_fourierChar (-L v w)).hasFDerivAt.comp w ha.neg\n  ext y\n  simp only [neg_mul, ContinuousLinearMap.coe_smul', ContinuousLinearMap.coe_comp', Pi.smul_apply,\n    Function.comp_apply, ofRealCLM_apply, smul_eq_mul, ContinuousLinearMap.comp_neg,\n    ContinuousLinearMap.neg_apply, ContinuousLinearMap.smulRight_apply,\n    ContinuousLinearMap.one_apply, real_smul, neg_inj]\n  ring\n\nlemma fderiv_fourierChar_neg_bilinear_right_apply (v : V) (w y : W) :\n    fderiv \u211d (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102)) w y = -2 * \u03c0 * I * L v y * \ud835\udc1e (-L v w) := by\n  simp [(hasFDerivAt_fourierChar_neg_bilinear_right L v w).fderiv]\n  ring\n\nlemma differentiable_fourierChar_neg_bilinear_right (v : V) :\n    Differentiable \u211d (fun w \u21a6 (\ud835\udc1e (-L v w) : \u2102)) :=\n  fun w \u21a6 (hasFDerivAt_fourierChar_neg_bilinear_right L v w).differentiableAt\n\nlemma hasFDerivAt_fourierChar_neg_bilinear_left (v : V) (w : W) :\n    HasFDerivAt (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102))\n      ((-2 * \u03c0 * I * \ud835\udc1e (-L v w)) \u2022 (ofRealCLM \u2218L (L.flip w))) v :=\n  hasFDerivAt_fourierChar_neg_bilinear_right L.flip w v\n\nlemma fderiv_fourierChar_neg_bilinear_left_apply (v y : V) (w : W) :\n    fderiv \u211d (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102)) v y = -2 * \u03c0 * I * L y w * \ud835\udc1e (-L v w) := by\n  simp [(hasFDerivAt_fourierChar_neg_bilinear_left L v w).fderiv]\n  ring\n\nlemma differentiable_fourierChar_neg_bilinear_left (w : W) :\n    Differentiable \u211d (fun v \u21a6 (\ud835\udc1e (-L v w) : \u2102)) :=\n  fun v \u21a6 (hasFDerivAt_fourierChar_neg_bilinear_left L v w).differentiableAt\n\nend Real\n\nvariable {E : Type*} [NormedAddCommGroup E] [NormedSpace \u2102 E]\n\nnamespace VectorFourier\n\nvariable {V W : Type*} [NormedAddCommGroup V] [NormedSpace \u211d V]\n  [NormedAddCommGroup W] [NormedSpace \u211d W] (L : V \u2192L[\u211d] W \u2192L[\u211d] \u211d) (f : V \u2192 E)\n\nvariable {f}\n\n/-- The Fourier integral of the derivative of a function is obtained by multiplying the Fourier\nintegral of the original function by `-L w v`. -/\ntheorem fourierIntegral_fderiv [MeasurableSpace V] [BorelSpace V] [FiniteDimensional \u211d V]\n    {\u03bc : Measure V} [Measure.IsAddHaarMeasure \u03bc]\n    (hf : Integrable f \u03bc) (h'f : Differentiable \u211d f) (hf' : Integrable (fderiv \u211d f) \u03bc) :\n    fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 (fderiv \u211d f)\n      = fourierSMulRight (-L.flip) (fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 f) := by\n  ext w y\n  let g : V \u2192 \u2102 := fun v \u21a6 \ud835\udc1e (-L v w)\n  have J : Integrable (fun v \u21a6 \ud835\udc1e (-(L v) w) \u2022 fderiv \u211d f v) \u03bc :=\n    (fourierIntegral_convergent_iff' _ _).2 hf'\n  /- First rewrite things in a simplified form, without any real change. -/\n  suffices \u222b x, g x \u2022 fderiv \u211d f x y \u2202\u03bc = \u222b x, (2 * \u2191\u03c0 * I * L y w * g x) \u2022 f x \u2202\u03bc by\n    simpa only [fourierIntegral, ContinuousLinearMap.toLinearMap\u2082_apply,\n      ContinuousLinearMap.integral_apply J, ContinuousLinearMap.coe_smul', Pi.smul_apply,\n      fourierSMulRight_apply, ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply, \u2190\n      integral_smul, neg_smul, smul_neg, \u2190 smul_smul, Complex.coe_smul, neg_neg]\n  have A x : fderiv \u211d g x y = - 2 * \u2191\u03c0 * I * L y w * g x :=\n    fderiv_fourierChar_neg_bilinear_left_apply _ _ _ _\n  /- Key step: integrate by parts with respect to `y` to switch the derivative from `f` to `g`. -/\n  rw [integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable, \u2190 integral_neg]\n  \u00b7 congr with x\n    simp only [A, neg_mul, neg_smul, neg_neg]\n  \u00b7 have : Integrable (fun x \u21a6 (-(2 * \u2191\u03c0 * I * \u2191((L y) w)) \u2022 ((g x : \u2102) \u2022 f x))) \u03bc :=\n      ((fourierIntegral_convergent_iff' _ _).2 hf).smul _\n    convert this using 2 with x\n    simp [A, smul_smul]\n  \u00b7 exact (fourierIntegral_convergent_iff' _ _).2 (hf'.apply_continuousLinearMap _)\n  \u00b7 exact (fourierIntegral_convergent_iff' _ _).2 hf\n  \u00b7 exact differentiable_fourierChar_neg_bilinear_left _ _\n  \u00b7 exact h'f\n\nopen ContinuousMultilinearMap\n\nvariable [SecondCountableTopology V] [MeasurableSpace V] [BorelSpace V] {\u03bc : Measure V}\n\n/-- The Fourier integral of the `n`-th derivative of a function is obtained by multiplying the\nFourier integral of the original function by `(2\u03c0I L w \u2b1d )^n`. -/\ntheorem fourierIntegral_iteratedFDeriv [FiniteDimensional \u211d V]\n    {\u03bc : Measure V} [Measure.IsAddHaarMeasure \u03bc] {N : \u2115\u221e} (hf : ContDiff \u211d N f)\n    (h'f : \u2200 (n : \u2115), n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f) \u03bc) {n : \u2115} (hn : n \u2264 N) :\n    fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 (iteratedFDeriv \u211d n f)\n      = (fun w \u21a6 fourierPowSMulRight (-L.flip) (fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 f) w n) := by\n  induction n with\n  | zero =>\n    ext w m\n    have I : Integrable (fun v \u21a6 \ud835\udc1e (- L v w) \u2022 iteratedFDeriv \u211d 0 f v) \u03bc :=\n      (fourierIntegral_convergent_iff' _ _).2 (h'f 0 bot_le)\n    simp only [Nat.zero_eq, fourierIntegral, ContinuousLinearMap.toLinearMap\u2082_apply,\n      integral_apply I, smul_apply, iteratedFDeriv_zero_apply, fourierPowSMulRight_apply, pow_zero,\n      Finset.univ_eq_empty, ContinuousLinearMap.neg_apply, ContinuousLinearMap.flip_apply,\n      Finset.prod_empty, one_smul]\n  | succ n ih =>\n    ext w m\n    -- instance on next line should not be necessary, but proof breaks down without it.\n    let NS : NormedSpace \u211d (V [\u00d7n]\u2192L[\u211d] E) := by infer_instance\n    have J : Integrable (fderiv \u211d (iteratedFDeriv \u211d n f)) \u03bc := by\n      specialize h'f (n + 1) hn\n      simp_rw [iteratedFDeriv_succ_eq_comp_left] at h'f\n      exact (LinearIsometryEquiv.integrable_comp_iff _).1 h'f\n    suffices H : (fourierIntegral \ud835\udc1e \u03bc L.toLinearMap\u2082 (fderiv \u211d (iteratedFDeriv \u211d n f)) w)\n          (m 0) (Fin.tail m) =\n        (-(2 * \u03c0 * I)) ^ (n + 1) \u2022 (\u220f x : Fin (n + 1), -L (m x) w) \u2022 \u222b v, \ud835\udc1e (-L v w) \u2022 f v \u2202\u03bc by\n      have A : \u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v (m 0)) (Fin.tail m) \u2202\u03bc\n          = (\u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v (m 0)) \u2202\u03bc) (Fin.tail m) := by\n        rw [integral_apply]\n        \u00b7 simp only [smul_apply]\n        \u00b7 exact (fourierIntegral_convergent_iff' L w).2 (J.apply_continuousLinearMap _)\n      have B : \u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v (m 0)) \u2202\u03bc =\n          (\u222b v, \ud835\udc1e (-L v w) \u2022 (fderiv \u211d (iteratedFDeriv \u211d n f) v) \u2202\u03bc) (m 0) := by\n        rw [ContinuousLinearMap.integral_apply]\n        \u00b7 simp only [ContinuousLinearMap.coe_smul', Pi.smul_apply]\n        \u00b7 exact (fourierIntegral_convergent_iff' L w).2 J\n      simp only [fourierIntegral, ContinuousLinearMap.toLinearMap\u2082_apply,\n        integral_apply ((fourierIntegral_convergent_iff' L w).2 (h'f _ hn)), smul_apply,\n        iteratedFDeriv_succ_apply_left, fourierPowSMulRight_apply, ContinuousLinearMap.neg_apply,\n        ContinuousLinearMap.flip_apply, A, B]\n      exact H\n    have h'n : n < N := lt_of_lt_of_le (by simp [-Nat.cast_succ]) hn\n    rw [fourierIntegral_fderiv]\n    \u00b7 have A : \u2200 (x : \u211d) (v : E), x \u2022 v = (x : \u2102) \u2022 v := fun x v \u21a6 rfl\n      simp only [ih h'n.le, fourierSMulRight_apply, ContinuousLinearMap.neg_apply,\n        ContinuousLinearMap.flip_apply, neg_smul, smul_neg, neg_neg, smul_apply,\n        fourierPowSMulRight_apply, A, smul_smul]\n      congr 1\n      have B : \u2200 (i : Fin n), Fin.tail m i = m (Fin.succ i) := fun i \u21a6 rfl\n      simp only [ofReal_prod, ofReal_neg, pow_succ, mul_neg, Fin.prod_univ_succ, neg_mul,\n        ofReal_mul, neg_neg, B]\n      ring\n    \u00b7 exact h'f n h'n.le\n    \u00b7 exact hf.differentiable_iteratedFDeriv h'n\n    \u00b7 exact J\n\nend VectorFourier\n\nnamespace Real\nopen VectorFourier\n\nvariable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace \u211d V] [FiniteDimensional \u211d V]\n  [MeasurableSpace V] [BorelSpace V] {f : V \u2192 E}\n\n/-- The Fourier integral of the Fr\u00e9chet derivative of a function is obtained by multiplying the\nFourier integral of the original function by `2\u03c0I \u27eav, w\u27eb`. -/\ntheorem fourierIntegral_fderiv\n    (hf : Integrable f) (h'f : Differentiable \u211d f) (hf' : Integrable (fderiv \u211d f)) :\n    \ud835\udcd5 (fderiv \u211d f) = fourierSMulRight (-innerSL \u211d) (\ud835\udcd5 f) := by\n  rw [\u2190 innerSL_real_flip V]\n  exact VectorFourier.fourierIntegral_fderiv (innerSL \u211d) hf h'f hf'\n\n/-- The Fourier integral of the `n`-th derivative of a function is obtained by multiplying the\nFourier integral of the original function by `(2\u03c0I L w \u2b1d )^n`. -/\ntheorem fourierIntegral_iteratedFDeriv {N : \u2115\u221e} (hf : ContDiff \u211d N f)\n    (h'f : \u2200 (n : \u2115), n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f)) {n : \u2115} (hn : n \u2264 N) :\n    \ud835\udcd5 (iteratedFDeriv \u211d n f)\n      = (fun w \u21a6 fourierPowSMulRight (-innerSL \u211d) (\ud835\udcd5 f) w n) := by\n  rw [\u2190 innerSL_real_flip V]\n  exact VectorFourier.fourierIntegral_iteratedFDeriv (innerSL \u211d) hf h'f hn\n\n/-- The Fourier integral of the Fr\u00e9chet derivative of a function is obtained by multiplying the\nFourier integral of the original function by `2\u03c0I x`. -/\ntheorem fourierIntegral_deriv\n    {f : \u211d \u2192 E} (hf : Integrable f) (h'f : Differentiable \u211d f) (hf' : Integrable (deriv f)) :\n    \ud835\udcd5 (deriv f) = fun (x : \u211d) \u21a6 (2 * \u03c0 * I * x) \u2022 (\ud835\udcd5 f x) := by\n  ext x\n  have I : Integrable (fun x \u21a6 fderiv \u211d f x) := by\n    simp_rw [\u2190 deriv_fderiv]\n    change Integrable (fun x \u21a6 ContinuousLinearMap.smulRightL _ _ _ 1 (deriv f x)) volume\n    apply ContinuousLinearMap.integrable_comp _ hf'\n  have : \ud835\udcd5 (deriv f) x = \ud835\udcd5 (fderiv \u211d f) x 1 := by\n    simp_rw [fourierIntegral_eq, deriv,\n      ContinuousLinearMap.integral_apply ((fourierIntegral_convergent_iff _).2 I)]\n    rfl\n  rw [this, fourierIntegral_fderiv hf h'f I]\n  have : x \u2022 \ud835\udcd5 f x = (x : \u2102) \u2022 \ud835\udcd5 f x := rfl\n  simp only [fourierSMulRight_apply, ContinuousLinearMap.neg_apply, innerSL_apply, smul_smul,\n    RCLike.inner_apply, conj_trivial, mul_one, neg_smul, smul_neg, neg_neg, neg_mul, this]\n\n", "theoremStatement": "theorem fourierIntegral_iteratedDeriv {f : \u211d \u2192 E} {N : \u2115\u221e} {n : \u2115} (hf : ContDiff \u211d N f)\n    (h'f : \u2200 (n : \u2115), n \u2264 N \u2192 Integrable (iteratedDeriv n f)) (hn : n \u2264 N) :\n    \ud835\udcd5 (iteratedDeriv n f) = fun (x : \u211d) \u21a6 (2 * \u03c0 * I * x) ^ n \u2022 (\ud835\udcd5 f x) ", "theoremName": "Real.fourierIntegral_iteratedDeriv", "fileCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "theoremCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv.jsonl", "positionMetadata": {"lineInFile": 279, "tokenPositionInFile": 14460, 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"Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Algebra.CharP.Invertible", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "Mathlib.Analysis.Fourier.FourierTransformDeriv"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  ext x : 1\n  have A : \u2200 (n : \u2115), n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f) := by\n    intro n hn\n    rw [iteratedFDeriv_eq_equiv_comp]\n    exact (LinearIsometryEquiv.integrable_comp_iff _).2 (h'f n hn)\n  have B : \ud835\udcd5 (fun x \u21a6 (iteratedFDeriv \u211d n f x) (fun i \u21a6 1)) x =\n      \ud835\udcd5 (iteratedFDeriv \u211d n f) x (fun i \u21a6 1) := by\n    rw [fourierIntegral_eq, fourierIntegral_eq, ContinuousMultilinearMap.integral_apply]\n    \u00b7 rfl\n    \u00b7 exact (fourierIntegral_convergent_iff _).2 (A n hn)\n  have C : \u2200 (c : \u211d) (v : E), c \u2022 v = (c : \u2102) \u2022 v := fun c v \u21a6 rfl\n  change \ud835\udcd5 (fun x \u21a6 iteratedDeriv n f x) x = _\n  simp_rw [iteratedDeriv, B, fourierIntegral_iteratedFDeriv hf A hn]\n  simp [C, smul_smul, \u2190 mul_pow]", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 695}}
+{"srcContext": "import Mathlib.Analysis.Distribution.SchwartzSpace\nimport Mathlib.MeasureTheory.Integral.IntegralEqImproper\nimport Mathlib.Topology.ContinuousFunction.Bounded\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.Inversion\n\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport PrimeNumberTheoremAnd.Sobolev\n\nopen FourierTransform Real Complex MeasureTheory Filter Topology BoundedContinuousFunction SchwartzMap VectorFourier BigOperators\n\nlocal instance {E : Type*} : Coe (E \u2192 \u211d) (E \u2192 \u2102) := \u27e8fun f n => f n\u27e9\n\nsection lemmas\n\n@[simp]\ntheorem nnnorm_eq_of_mem_circle (z : circle) : \u2016z.val\u2016\u208a = 1 := NNReal.coe_eq_one.mp (by simp)\n\n@[simp]\ntheorem nnnorm_circle_smul (z : circle) (s : \u2102) : \u2016z \u2022 s\u2016\u208a = \u2016s\u2016\u208a := by\n  simp [show z \u2022 s = z.val * s from rfl]\n\nnoncomputable def e (u : \u211d) : \u211d \u2192\u1d47 \u2102 where\n  toFun v := \ud835\udc1e (-v * u)\n  map_bounded' := \u27e82, fun x y => (dist_le_norm_add_norm _ _).trans (by simp [one_add_one_eq_two])\u27e9\n\n@[simp] lemma e_apply (u : \u211d) (v : \u211d) : e u v = \ud835\udc1e (-v * u) := rfl\n\ntheorem hasDerivAt_e {u x : \u211d} : HasDerivAt (e u) (-2 * \u03c0 * u * I * e u x) x := by\n  have l2 : HasDerivAt (fun v => -v * u) (-u) x := by simpa only [neg_mul_comm] using hasDerivAt_mul_const (-u)\n  convert (hasDerivAt_fourierChar (-x * u)).scomp x l2 using 1\n  simp ; ring\n\nlemma fourierIntegral_deriv_aux2 (e : \u211d \u2192\u1d47 \u2102) {f : \u211d \u2192 \u2102} (hf : Integrable f) : Integrable (\u21d1e * f) :=\n  hf.bdd_mul e.continuous.aestronglyMeasurable \u27e8_, e.norm_coe_le_norm\u27e9\n\n@[simp] lemma F_neg {f : \u211d \u2192 \u2102} {u : \u211d} : \ud835\udcd5 (fun x => -f x) u = - \ud835\udcd5 f u := by\n  simp [fourierIntegral_eq, integral_neg]\n\n@[simp] lemma F_add {f g : \u211d \u2192 \u2102} (hf : Integrable f) (hg : Integrable g) (x : \u211d) :\n    \ud835\udcd5 (fun x => f x + g x) x = \ud835\udcd5 f x + \ud835\udcd5 g x :=\n  congr_fun (fourierIntegral_add continuous_fourierChar (by exact continuous_mul) hf hg).symm x\n\n@[simp] lemma F_sub {f g : \u211d \u2192 \u2102} (hf : Integrable f) (hg : Integrable g) (x : \u211d) :\n    \ud835\udcd5 (fun x => f x - g x) x = \ud835\udcd5 f x - \ud835\udcd5 g x := by\n  simp_rw [sub_eq_add_neg] ; rw [F_add] ; simp ; exact hf ; exact hg.neg\n\n@[simp] lemma F_mul {f : \u211d \u2192 \u2102} {c : \u2102} {u : \u211d} : \ud835\udcd5 (fun x => c * f x) u = c * \ud835\udcd5 f u := by\n  simp [fourierIntegral_real_eq, \u2190 integral_mul_left] ; congr ; ext\n  simp [Real.fourierChar, expMapCircle] ; ring\n\nend lemmas\n\ntheorem fourierIntegral_self_add_deriv_deriv (f : W21) (u : \u211d) :\n    (1 + u ^ 2) * \ud835\udcd5 f u = \ud835\udcd5 (fun u => f u - (1 / (4 * \u03c0 ^ 2)) * deriv^[2] f u) u := by\n  have l1 : Integrable (fun x => (((\u03c0 : \u2102) ^ 2)\u207b\u00b9 * 4\u207b\u00b9) * deriv (deriv f) x) := by\n    apply Integrable.const_mul ; simpa [iteratedDeriv_succ] using f.integrable le_rfl\n  have l4 : Differentiable \u211d f := f.differentiable\n  have l5 : Differentiable \u211d (deriv f) := f.deriv.differentiable\n  simp [f.hf, l1, add_mul, Real.fourierIntegral_deriv f.hf' l5 f.hf'', Real.fourierIntegral_deriv f.hf l4 f.hf']\n  field_simp [pi_ne_zero] ; ring_nf ; simp\n\n@[simp] lemma deriv_ofReal : deriv ofReal' = fun _ => 1 := by\n  ext x ; exact ((hasDerivAt_id x).ofReal_comp).deriv\n\ntheorem bla (a : \u2102) (f : \u211d \u2192 \u2102) (n : \u2115) (hf : ContDiff \u211d n f) :\n    iteratedDeriv n (fun x \u21a6 a * x * f x) = fun x =>\n      a * x * iteratedDeriv n f x + n * a * iteratedDeriv (n - 1) f x := by\n\n  induction n with\n  | zero => simp\n  | succ n ih =>\n    have l0 : ContDiff \u211d n f := hf.of_succ\n    rw [iteratedDeriv_succ, ih l0] ; ext x\n    have l5 : ContDiff \u211d (\u2191(1 + n)) f := by convert hf using 1 ; simp ; ring\n    have l4 : DifferentiableAt \u211d (fun x \u21a6 iteratedDeriv n f x) x := by\n      have := ((l5.iterate_deriv' 1 n).differentiable le_rfl).differentiableAt (x := x)\n      simpa [iteratedDeriv_eq_iterate] using this\n    have l3 : DifferentiableAt \u211d (fun x \u21a6 a * \u2191x) x := by\n      apply DifferentiableAt.const_mul\n      exact (contDiff_ofReal.differentiable le_top).differentiableAt\n    have l1 : DifferentiableAt \u211d (fun x \u21a6 a * \u2191x * iteratedDeriv n f x) x := l3.mul l4\n    have l2 : DifferentiableAt \u211d (fun x \u21a6 \u2191n * a * iteratedDeriv (n - 1) f x) x := by\n      apply DifferentiableAt.const_mul\n      apply l5.differentiable_iteratedDeriv\n      norm_cast ; exact Nat.sub_le _ _ |>.trans_lt (by simp)\n    simp [deriv_add l1 l2, deriv_mul l3 l4, \u2190 iteratedDeriv_succ]\n    cases n <;> simp <;> ring\n\nnoncomputable def MS (a : \u2102) (f : \ud835\udce2(\u211d, \u2102)) : \ud835\udce2(\u211d, \u2102) where\n  toFun x := a * x * f x\n  smooth' := contDiff_const.mul contDiff_ofReal |>.mul f.smooth'\n  decay' k n := by\n    simp only [norm_iteratedFDeriv_eq_norm_iteratedDeriv]\n    simp_rw [bla a f n <| f.smooth'.of_le le_top]\n    obtain \u27e8C\u2081, hC\u2081\u27e9 := f.decay' (k + 1) n\n    obtain \u27e8C\u2082, hC\u2082\u27e9 := f.decay' k (n - 1)\n    use \u2016a\u2016 * C\u2081 + \u2016a\u2016 * n * C\u2082 ; intro x\n    have l2 := norm_add_le (a * x * iteratedDeriv n f x) (n * a * iteratedDeriv (n - 1) f x)\n    have l3 : 0 \u2264 \u2016x\u2016 ^ k := by positivity\n    apply (mul_le_mul_of_nonneg_left l2 l3).trans ; rw [mul_add] ; apply add_le_add\n    \u00b7 have : 0 \u2264 \u2016a\u2016 := by positivity\n      convert mul_le_mul_of_nonneg_left (hC\u2081 x) this using 1\n      simp [norm_iteratedFDeriv_eq_norm_iteratedDeriv, abs_eq_self.mpr pi_nonneg] ; ring_nf ; rfl\n    \u00b7 have : 0 \u2264 \u2016a\u2016 * n := by positivity\n      convert mul_le_mul_of_nonneg_left (hC\u2082 x) this using 1\n      simp [norm_iteratedFDeriv_eq_norm_iteratedDeriv, abs_eq_self.mpr pi_nonneg] ; ring_nf ; rfl\n\n@[simp] lemma MS_apply (a : \u2102) (f : \ud835\udce2(\u211d, \u2102)) (x : \u211d) : MS a f x = (a * x) \u2022 f x := rfl\n\nlemma MS_iterate (a : \u2102) (f : \ud835\udce2(\u211d, \u2102)) (n : \u2115) : (MS a)^[n] f = fun x : \u211d => (a * x) ^ n \u2022 f x := by\n  induction n generalizing f with\n  | zero => simp\n  | succ n ih => ext x ; simp [ih, pow_succ] ; ring\n\nlemma fourierIntegral_decay_aux (f : \u211d \u2192 \u2102) (k : \u2115) (h1 : ContDiff \u211d k f)\n    (h2 : \u2200 n \u2264 k, Integrable (iteratedDeriv n f)) (x : \u211d) :\n    \u2016(2 * \u03c0 * I * x) ^ k \u2022 \ud835\udcd5 f x\u2016 \u2264 (\u222b y : \u211d, \u2016iteratedDeriv k f y\u2016) := by\n  have l2 (x : \u211d) : (2 * \u03c0 * I * x) ^ k \u2022 \ud835\udcd5 f x = \ud835\udcd5 (iteratedDeriv k f) x := by\n    simp [Real.fourierIntegral_iteratedDeriv h1 (fun n hn => h2 n <| Nat.cast_le.mp hn) le_rfl]\n  simpa only [l2] using Fourier.norm_fourierIntegral_le_integral_norm ..\n\nlemma iteratedDeriv_schwartz (f : \ud835\udce2(\u211d, \u2102)) (n : \u2115) : iteratedDeriv n f = (SchwartzMap.derivCLM \u211d)^[n] f := by\n  induction n with\n  | zero => rfl\n  | succ n ih => rw [iteratedDeriv_succ, ih, Function.iterate_succ'] ; rfl\n\ntheorem fourierIntegral_decay (f : \ud835\udce2(\u211d, \u2102)) (k : \u2115) : \u2203 C, \u2200 (x : \u211d), \u2016x\u2016 ^ k * \u2016\ud835\udcd5 f x\u2016 \u2264 C := by\n  convert_to \u2203 C, \u2200 x : \u211d, \u2016x ^ k * \ud835\udcd5 f x\u2016 \u2264 C ; \u00b7 simp\n  convert_to \u2203 C, \u2200 x : \u211d, \u2016(2 * \u03c0 * I * x) ^ k * \ud835\udcd5 f x\u2016 / (2 * \u03c0) ^ k \u2264 C using 4\n  \u00b7 field_simp [mul_pow, abs_eq_self.mpr pi_nonneg] ; ring\n  convert_to \u2203 C, \u2200 x : \u211d, \u2016(2 * \u03c0 * I * x) ^ k \u2022 \ud835\udcd5 f x\u2016 / (2 * \u03c0) ^ k \u2264 C\n  use (\u222b (y : \u211d), \u2016iteratedDeriv k (\u21d1f) y\u2016) / (2 * \u03c0) ^ k ; intro x\n  have l1 : \u2200 n \u2264 k, Integrable (iteratedDeriv n f) volume := by\n    simp_rw [iteratedDeriv_schwartz] ; simp [SchwartzMap.integrable]\n  have := fourierIntegral_decay_aux f k (f.smooth'.of_le le_top) l1 x\n  apply div_le_div_of_nonneg_right this (by positivity)\n\nnoncomputable def FS (f : \ud835\udce2(\u211d, \u2102)) : \ud835\udce2(\u211d, \u2102) where\n  toFun := \ud835\udcd5 f\n  smooth' := by\n    rw [contDiff_top] ; intro n\n    apply Real.contDiff_fourierIntegral ; intro k _\n    apply SchwartzMap.integrable_pow_mul\n  decay' := by\n    simp only [norm_iteratedFDeriv_eq_norm_iteratedDeriv]\n    intro k n\n    have l1 (k : \u2115) (_ : k \u2264 (n : \u2115\u221e)) : Integrable (fun x \u21a6 x ^ k \u2022 f x) volume := by\n      convert_to Integrable ((MS 1)^[k] f) ; \u00b7 simp [MS_iterate]\n      apply SchwartzMap.integrable\n    simp_rw [@Real.iteratedDeriv_fourierIntegral \u2102 _ _ f n n l1 le_rfl]\n    convert_to \u2203 C, \u2200 (x : \u211d), \u2016x\u2016 ^ k * \u2016\ud835\udcd5 ((MS (-2 * \u03c0 * I))^[n] f) x\u2016 \u2264 C ; \u00b7 simp [MS_iterate]\n    apply fourierIntegral_decay\n\n@[simp] lemma FS_apply (f : \ud835\udce2(\u211d, \u2102)) (x : \u211d) : FS f x = \ud835\udcd5 f x := rfl\n\n", "theoremStatement": "@[simp] lemma FS_toFun (f : \ud835\udce2(\u211d, \u2102)) : \u21d1(FS f) = \ud835\udcd5 f ", "theoremName": "FS_toFun", "fileCreated": {"commit": "00a2f4cf69f1409c26fcd9d398e985f4b733128e", "date": "2024-03-20"}, "theoremCreated": {"commit": "53de16ec5814caedef08dce9b306b92924a08717", "date": "2024-04-08"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Fourier.lean", "module": "PrimeNumberTheoremAnd.Fourier", "jsonFile": "PrimeNumberTheoremAnd.Fourier.jsonl", "positionMetadata": {"lineInFile": 157, "tokenPositionInFile": 7632, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": 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"Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\n", "theoremStatement": "lemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) ", "theoremName": "hh_integrable_aux", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "7d53759dfc8d08e60e519bb2843d30571f54df85", "date": "2024-03-27"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1264, "tokenPositionInFile": 62776, "theoremPositionInFile": 116}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 445, 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"Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring", "proofType": "tactic", "proofLengthLines": 71, "proofLengthTokens": 2842}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Basic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.Wiener\n\nopen Asymptotics Complex ComplexConjugate Topology Filter Real MeasureTheory Set\n\nopen scoped Interval\n\n-- TODO: why do we need to bump this?\ninstance : MeasurableDiv\u2082 \u211d := by\n  haveI (G : Type) [DivInvMonoid G] [MeasurableSpace G] [MeasurableInv G] [MeasurableMul\u2082 G] :\n    MeasurableDiv\u2082 G := inferInstance\n  exact this \u211d\n\n/-%%\nIn this section, we prove the Perron formula, which plays a key role in our proof of Mellin inversion.\n%%-/\n\n/-%%\nThe following is preparatory material used in the proof of the Perron formula, see Lemma \\ref{formulaLtOne}.\n%%-/\n\n/-%\nTODO: move to general section.\n\\begin{lemma}[zeroTendstoDiff]\\label{zeroTendstoDiff}\\lean{zeroTendstoDiff}\\leanok\nIf the limit of $0$ is $L\u2081 - L\u2082$, then $L\u2081 = L\u2082$.\n\\end{lemma}\n%-/\nlemma zeroTendstoDiff (L\u2081 L\u2082 : \u2102) (f : \u211d \u2192 \u2102) (h : \u2200\u1da0 T in atTop,  f T = 0)\n    (h' : Tendsto f atTop (\ud835\udcdd (L\u2082 - L\u2081))) : L\u2081 = L\u2082 := by\n  rw [\u2190 zero_add L\u2081, \u2190 @eq_sub_iff_add_eq]\n  exact tendsto_nhds_unique (EventuallyEq.tendsto h) h'\n/-%\n\\begin{proof}\\leanok\nObvious.\n\\end{proof}\n%-/\n\n/-%\nTODO: Move this to general section.\n\\begin{lemma}[RectangleIntegral_tendsTo_VerticalIntegral]\\label{RectangleIntegral_tendsTo_VerticalIntegral}\\lean{RectangleIntegral_tendsTo_VerticalIntegral}\\leanok\n\\uses{RectangleIntegral}\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\lim_{T\\to\\infty}\\int_{\\sigma-iT}^{\\sigma'+iT}f(s)ds =\n\\int_{(\\sigma')}f(s)ds - \\int_{(\\sigma)}f(s)ds.$$\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_VerticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (T : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) atTop\n      (\ud835\udcdd (VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3)) := by\n/-%\n\\begin{proof}\\leanok\nAlmost by definition.\n%-/\n  simp only [RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im, ofReal_im,\n    mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add, zero_sub,\n    add_im]\n  apply Tendsto.sub\n  \u00b7 rewrite [\u2190 zero_add (VerticalIntegral _ _), \u2190 zero_sub_zero]\n    apply Tendsto.add <| Tendsto.sub (hbot.comp tendsto_neg_atTop_atBot) htop\n    exact (intervalIntegral_tendsto_integral hright tendsto_neg_atTop_atBot tendsto_id).const_smul I\n  \u00b7 exact (intervalIntegral_tendsto_integral hleft tendsto_neg_atTop_atBot tendsto_id).const_smul I\n--%\\end{proof}\n\nlemma verticalIntegral_eq_verticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hf : HolomorphicOn f ([[\u03c3,  \u03c3']] \u00d7\u2102 univ))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    VerticalIntegral f \u03c3 = VerticalIntegral f \u03c3' := by\n  refine zeroTendstoDiff _ _ _ (univ_mem' fun _ \u21a6 ?_)\n    (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  exact integral_boundary_rect_eq_zero_of_differentiableOn f _ _\n    (hf.mono fun z hrect \u21a6 \u27e8by simpa using hrect.1, trivial\u27e9)\n\nlemma verticalIntegral_sub_verticalIntegral_eq_squareIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102} {p : \u2102}\n    (h\u03c3: \u03c3 < p.re \u2227 p.re < \u03c3') (hf : HolomorphicOn f (Icc \u03c3  \u03c3' \u00d7\u2102 univ \\ {p}))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3 =\n    RectangleIntegral f (-c - c * I + p) (c + c * I + p) := by\n  have : Icc \u03c3 \u03c3' \u00d7\u2102 univ \u2208 \ud835\udcdd p := by\n    rw [\u2190 mem_interior_iff_mem_nhds, Complex.interior_reProdIm, interior_Icc, interior_univ]\n    refine \u27e8\u27e8?_, ?_\u27e9, trivial\u27e9 <;> linarith\n  obtain \u27e8c', hc'0, hc'\u27e9 := ((nhds_hasBasis_square p).1 _).mp this\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' hc'0] with c \u27e8hc0, hcc'\u27e9\n  have hsub : Square p c \u2286 Icc \u03c3 \u03c3' \u00d7\u2102 univ := (square_subset_square hc0 hcc'.le).trans hc'\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  apply Filter.EventuallyEq.tendsto\n  filter_upwards [Filter.Ioi_mem_atTop ((c - p.im) \u2294 (c + p.im))] with y hy\n  have : c - p.im < y \u2227 c + p.im < y := sup_lt_iff.mp hy\n  have : c + \u03c3 \u2264 p.re := by simpa using (hsub \u27e8left_mem_uIcc, left_mem_uIcc\u27e9).1.1\n  have : c + p.re \u2264 \u03c3' := by simpa using (hsub \u27e8right_mem_uIcc, right_mem_uIcc\u27e9).1.2\n  apply RectanglePullToNhdOfPole'\n  \u00b7 simpa using \u27e8by linarith, by linarith, by linarith\u27e9\n  \u00b7 exact square_mem_nhds p (ne_of_gt hc0)\n  \u00b7 apply RectSubRect' <;> simpa using by linarith\n  \u00b7 refine hf.mono (diff_subset_diff ?_ subset_rfl)\n    simpa [Rectangle, uIcc_of_lt (h\u03c3.1.trans h\u03c3.2)] using fun x \u27e8hx, _\u27e9 \u21a6 \u27e8hx, trivial\u27e9\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_UpperU]\\label{RectangleIntegral_tendsTo_UpperU}\\lean{RectangleIntegral_tendsTo_UpperU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma+iT}^{\\sigma'+iU}f(s)ds$$\nas $U\\to\\infty$ is the ``UpperUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_UpperU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 + I * T) (\u03c3' + I * U)) atTop\n      (\ud835\udcdd (UpperUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, UpperUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  + I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  + I * t).im = t  := by simp\n  have hbot : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) := by\n    exact tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in T..U, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Ioi T, f (s + y * I)) := by\n    exact (intervalIntegral_tendsto_integral_Ioi T int.restrict tendsto_id).const_smul I\n  have := ((hbot.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  simpa only [RectangleIntegral, UpperUIntegral, h_re, h_im, sub_zero, \u2190integral_Ici_eq_integral_Ioi]\n--%\\end{proof}\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_LowerU]\\label{RectangleIntegral_tendsTo_LowerU}\\lean{RectangleIntegral_tendsTo_LowerU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to -\\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma-iU}^{\\sigma'-iT}f(s)ds$$\nas $U\\to\\infty$ is the ``LowerUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_LowerU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * U) (\u03c3' - I * T)) atTop\n      (\ud835\udcdd (- LowerUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, LowerUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  - I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  - I * t).im = -t  := by simp\n  have hbot' : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - y * I)) atTop (\ud835\udcdd 0) := by\n    convert (hbot.comp tendsto_neg_atTop_atBot) using 1\n    ext; simp only [Function.comp_apply, ofReal_neg, neg_mul]; rfl\n  have htop : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) :=\n    tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in -U..-T, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Iic (-T), f (s + y * I)) := by\n    have := (intervalIntegral_tendsto_integral_Iic (-T) int.restrict tendsto_id).const_smul I\n    convert (this.comp tendsto_neg_atTop_atBot) using 1\n  have := ((hbot'.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  rw [zero_sub] at this\n  simp_rw [RectangleIntegral, LowerUIntegral, HIntegral, VIntegral, h_re, h_im, ofReal_neg, neg_mul, neg_add_rev, neg_sub]\n  have final : (((-\u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I)) + I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) -\n      I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) = (-(I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) +\n      ((I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) - \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I))) := by\n    ring_nf\n  exact final \u25b8 this\n--%\\end{proof}\n\n/-%%\nTODO : Move to general section\n\\begin{lemma}[limitOfConstant]\\label{limitOfConstant}\\lean{limitOfConstant}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma>0$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstant {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : 0 < \u03c3') (_ : 0 < \u03c3''), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atTop (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\\begin{align*}\n\\lim_{\\sigma'\\to\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atTop \u03c3) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 (\u03c3pos.trans_le h) \u03c3pos\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[limitOfConstantLeft]\\label{limitOfConstantLeft}\\lean{limitOfConstantLeft}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma<-3/2$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to-\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstantLeft {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3lt : \u03c3 \u2264 -3/2)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : \u03c3' \u2264 -3/2) (_ : \u03c3'' \u2264 -3/2), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atBot (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\begin{align*}\n\\lim_{\\sigma'\\to-\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to-\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atBot (-3/2)) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 h \u03c3lt\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_lt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\leanok\nLet $x>0$ and $x<1$. Then\n$$\\lim_{\\sigma\\to\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_lt_one {x : \u211d}  (xpos : 0 < x) (x_lt_one : x < 1) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nStandard.\n%%-/\n  have := Tendsto.mul_const C (tendsto_rpow_atTop_of_base_lt_one x (by linarith) x_lt_one)\n  simpa only [rpow_eq_pow, zero_mul] using this\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_gt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\leanok\nLet $x>1$. Then\n$$\\lim_{\\sigma\\to-\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_gt_one {x : \u211d} (x_gt_one : 1 < x) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atBot (\ud835\udcdd 0) := by\n  have := (zero_lt_one.trans x_gt_one)\n  have h := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one (inv_pos.mpr this) (inv_lt_one x_gt_one) C\n  convert (h.comp tendsto_neg_atBot_atTop) using 1\n  ext; simp only [this.le, inv_rpow, Function.comp_apply, rpow_neg, inv_inv]\n\n/-%%\n\\begin{proof}\\leanok\nStandard.\n\\end{proof}\n%%-/\n\n-- TODO: move near `Complex.cpow_neg`?\nlemma Complex.cpow_inv_ofReal_pos {a : \u211d} (ha : 0 \u2264 a) (r : \u2102) :\n    ((a : \u2102) ^ r)\u207b\u00b9 = (a : \u2102)\u207b\u00b9 ^ r := by\n  sorry\n\nlemma Complex.cpow_eq_exp_log_ofReal (x : \u211d) (hx : 0 < x) (y : \u2102) :\n    (x : \u2102) ^ y = Complex.exp (Real.log x * y) := by\n  simp [\u2190 Complex.cpow_eq_pow, Complex.cpow, hx.ne.symm, \u2190 Complex.ofReal_log hx.le]\n\n-- TODO: move near `Complex.mul_cpow_ofReal_nonneg`\nlemma Complex.cpow_neg_eq_inv_pow_ofReal_pos {a : \u211d} (ha : 0 < a) (r : \u2102) :\n    (a : \u2102) ^ (-r) = (a\u207b\u00b9 : \u2102) ^ r := by\n  rw [cpow_neg, \u2190 Complex.inv_cpow]\n  exact slitPlane_arg_ne_pi (Or.inl ha)\n\nnamespace Perron\n\nvariable {x \u03c3 \u03c3' \u03c3'' T : \u211d}\n\nnoncomputable abbrev f (x : \u211d) := fun (s : \u2102) \u21a6 x ^ s / (s * (s + 1))\n\n\nlemma f_mul_eq_f {x t : \u211d} (tpos : 0 < t) (xpos : 0 < x) (s : \u2102) : f t s * (x : \u2102) ^ (-s) = f (t / x) s := by\n  by_cases s_eq_zero : s = 0\n  \u00b7 simp [f, s_eq_zero]\n  by_cases s_eq_neg_one : s = -1\n  \u00b7 simp [f, s_eq_neg_one]\n  field_simp [f, mul_ne_zero s_eq_zero (fun hs \u21a6 add_eq_zero_iff_eq_neg.mp hs |> s_eq_neg_one)]\n  convert (Complex.mul_cpow_ofReal_nonneg tpos.le (inv_pos.mpr xpos).le s).symm using 2\n  \u00b7 convert Complex.cpow_neg_eq_inv_pow_ofReal_pos xpos s\n    exact ofReal_inv x\n  \u00b7 simp only [ofReal_inv]; rfl\n\n/-%%\n\\begin{lemma}[isHolomorphicOn]\\label{isHolomorphicOn}\\lean{Perron.isHolomorphicOn}\\leanok\nLet $x>0$. Then the function $f(s) = x^s/(s(s+1))$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n\\end{lemma}\n%%-/\nlemma isHolomorphicOn (xpos : 0 < x) : HolomorphicOn (f x) {0, -1}\u1d9c := by\n/-%%\n\\begin{proof}\\leanok\nComposition of differentiabilities.\n%%-/\n  unfold f\n  simp_rw [Complex.cpow_def_of_ne_zero <| ofReal_ne_zero.mpr <| ne_of_gt xpos]\n  apply DifferentiableOn.div <| DifferentiableOn.cexp <| DifferentiableOn.const_mul differentiableOn_id _\n  \u00b7 exact DifferentiableOn.mul differentiableOn_id <| DifferentiableOn.add_const differentiableOn_id 1\n  \u00b7 intro x hx\n    obtain \u27e8h0, h1\u27e9 := not_or.mp hx\n    exact mul_ne_zero h0 <| add_ne_add_left 1 |>.mpr h1 |>.trans_eq (add_left_neg 1)\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[integralPosAux]\\label{integralPosAux}\\lean{Perron.integralPosAux}\\leanok\nThe integral\n$$\\int_\\R\\frac{1}{|(1+t^2)(2+t^2)|^{1/2}}dt$$\nis positive (and hence convergent - since a divergent integral is zero in Lean, by definition).\n\\end{lemma}\n%%-/\n\n", "theoremStatement": "lemma integral_one_div_const_add_sq_pos (c : \u211d) (hc : 0 < c) : 0 < \u222b (t : \u211d), 1 / (c + t ^ 2) ", "theoremName": "Perron.integral_one_div_const_add_sq_pos", "fileCreated": {"commit": "70815e04f04e213b27a1f0756724b06c69b56da1", "date": "2024-02-06"}, "theoremCreated": {"commit": "4651fb5df36fad9ba85f0ce00cb159d66a4632f4", "date": "2024-02-11"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/PerronFormula.lean", "module": "PrimeNumberTheoremAnd.PerronFormula", "jsonFile": "PrimeNumberTheoremAnd.PerronFormula.jsonl", "positionMetadata": {"lineInFile": 322, "tokenPositionInFile": 15073, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 187, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Integral.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have hfun_eq (t : \u211d) : 1 / (c + t ^ 2) = c\u207b\u00b9 * (1 + (c.sqrt\u207b\u00b9 * t) ^ 2)\u207b\u00b9 := by\n    field_simp [hc.ne.symm]\n  simp_rw [hfun_eq, MeasureTheory.integral_mul_left,\n    Measure.integral_comp_mul_left (fun t \u21a6 (1 + t ^ 2)\u207b\u00b9) (a:=c.sqrt\u207b\u00b9)]\n  simp [abs_eq_self.mpr <| Real.sqrt_nonneg c,\n    mul_pos (inv_pos.mpr hc) <| mul_pos (sqrt_pos.mpr hc) Real.pi_pos]", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 360}}
+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\ntheorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) := by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz\n\ntheorem primesBetween_le_siftedSum_add :\n    primesBetween x (x+y) \u2264 (primeInterSieve x y z hz).siftedSum + z := by\n  classical\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a (Finset.Icc 1 (Nat.floor z))).card\n  \u00b7 rw[primesBetween]\n    norm_cast\n    apply Finset.card_le_card\n    apply primesBetween_subset _ _ _ hx\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card\n    + \u2191(Finset.Icc 1 (Nat.floor z)).card\n  \u00b7 norm_cast\n    apply Finset.card_union_le\n  rw[siftedSum_eq_card]\n  gcongr\n  rw[Nat.card_Icc]\n  simp\n  apply Nat.floor_le\n  linarith\n\nsection Remainder\n\ntheorem Ioc_filter_dvd_eq (d a b: \u2115) (hd : d \u2260 0) :\n  Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b) =\n    Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d)) := by\n  ext n\n  simp only [Finset.mem_filter, Finset.mem_Ioc, Nat.ceil_le, Finset.mem_image,\n    Nat.le_floor_iff (show 0 \u2264 x+y by linarith)]\n  constructor\n  \u00b7 intro hn\n    use  n/d\n    rcases hn with \u27e8\u27e8han, hnb\u27e9, hd\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 exact Nat.div_lt_div_of_lt_of_dvd hd han\n    \u00b7 exact Nat.div_le_div_right (Nat.le_floor hnb)\n    \u00b7 exact Nat.div_mul_cancel hd\n  \u00b7 rintro \u27e8r, \u27e8ha, ha'\u27e9, rfl\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 refine (Nat.div_lt_iff_lt_mul ?_).mp ha\n      omega\n    \u00b7 exact Nat.mul_le_of_le_div d r b ha'\n    \u00b7 exact Nat.dvd_mul_left d r\n\ntheorem card_Ioc_filter_dvd (d a b: \u2115) (hd : d \u2260 0) :\n    (Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b)).card = b / d - a / d  := by\n  rw [Ioc_filter_dvd_eq _ _ _ hd]\n  rw [Finset.card_image_of_injective _ <| mul_left_injective\u2080 hd]\n  simp\n\n", "theoremStatement": "theorem multSum_eq (d : \u2115) (hd : d \u2260 0):\n    (primeInterSieve x y z hz).multSum d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) ", "theoremName": "BrunTitchmarsh.multSum_eq", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": 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"Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", 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"Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  unfold Sieve.multSum\n  rw[primeInterSieve]\n  simp\n  trans \u2191(Finset.Ioc (Nat.ceil x - 1) (Nat.floor (x+y)) |>.filter (d \u2223 \u00b7) |>.card)\n  \u00b7 rw [\u2190Nat.Icc_succ_left]\n    congr\n    rw [\u2190Nat.succ_sub]; rfl\n    simp [hx]\n  \u00b7 rw[BrunTitchmarsh.card_Ioc_filter_dvd _ _ _ hd]", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 272}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module sieve\n-/\nimport Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults\n\nnoncomputable section\n\nopen scoped BigOperators ArithmeticFunction\n\nopen Finset Real Nat Aux\n\nstructure Sieve where mk ::\n  support : Finset \u2115\n  prodPrimes : \u2115\n  prodPrimes_squarefree : Squarefree prodPrimes\n  weights : \u2115 \u2192 \u211d\n  weights_nonneg : \u2200 n : \u2115, 0 \u2264 weights n\n  totalMass : \u211d\n  nu : ArithmeticFunction \u211d\n  nu_mult : nu.IsMultiplicative\n  nu_pos_of_prime : \u2200 p : \u2115, p.Prime \u2192 p \u2223 prodPrimes \u2192 0 < nu p\n  nu_lt_one_of_prime : \u2200 p : \u2115, p.Prime \u2192 p \u2223 prodPrimes \u2192 nu p < 1\n\nattribute [arith_mult] Sieve.nu_mult\n\nnamespace Sieve\n\nvariable (s : Sieve)\nlocal notation3 \"\u03bd\" => Sieve.nu s\nlocal notation3 \"P\" => Sieve.prodPrimes s\nlocal notation3 \"a\" => Sieve.weights s\nlocal notation3 \"X\" => Sieve.totalMass s\nlocal notation3 \"A\" => Sieve.support s\n\n@[simp]\ndef multSum (d : \u2115) : \u211d :=\n  \u2211 n in A, if d \u2223 n then a n else 0\n\nlocal notation3 \"\ud835\udc9c\" => Sieve.multSum s\n\n-- A_d = \u03bd (d)/d X + R_d\n@[simp]\ndef rem (d : \u2115) : \u211d :=\n  \ud835\udc9c d - \u03bd d * X\n\nlocal notation3 \"R\" => Sieve.rem s\n\ndef siftedSum : \u211d :=\n  \u2211 d in A, if Coprime P d then a d else 0\n\n-- S = \u2211_{l|P, l\u2264\u221ay} g(l)\n-- Used in statement of the simple form of the selberg bound\ndef selbergTerms : ArithmeticFunction \u211d :=\n  s.nu.pmul (.prodPrimeFactors fun p =>  1 / (1 - \u03bd p))\n\nlocal notation3 \"g\" => Sieve.selbergTerms s\n\ndef selbergTerms_apply (d : \u2115) :\n    g d = \u03bd d * \u220f p in d.primeFactors, 1/(1 - \u03bd p) := by\n  unfold selbergTerms\n  by_cases h : d=0\n  \u00b7 rw [h]; simp\n  rw [ArithmeticFunction.pmul_apply, ArithmeticFunction.prodPrimeFactors_apply h]\n\ndef mainSum (\u03bcPlus : \u2115 \u2192 \u211d) : \u211d :=\n  \u2211 d in divisors P, \u03bcPlus d * \u03bd d\n\ndef errSum (\u03bcPlus : \u2115 \u2192 \u211d) : \u211d :=\n  \u2211 d in divisors P, |\u03bcPlus d| * |R d|\n\n\nsection UpperBoundSieve\n\ndef UpperMoebius (\u03bc_plus : \u2115 \u2192 \u211d) : Prop :=\n  \u2200 n : \u2115, (if n=1 then 1 else 0) \u2264 \u2211 d in n.divisors, \u03bc_plus d\n\nstructure UpperBoundSieve where mk ::\n  \u03bcPlus : \u2115 \u2192 \u211d\n  h\u03bcPlus : UpperMoebius \u03bcPlus\n\ninstance ubTo\u03bcPlus : CoeFun UpperBoundSieve fun _ => \u2115 \u2192 \u211d where coe ub := ub.\u03bcPlus\n\ndef LowerMoebius (\u03bcMinus : \u2115 \u2192 \u211d) : Prop :=\n  \u2200 n : \u2115, \u2211 d in n.divisors, \u03bcMinus d \u2264 (if n=1 then 1 else 0)\n\nstructure LowerBoundSieve where mk ::\n  \u03bcMinus : \u2115 \u2192 \u211d\n  h\u03bcMinus : LowerMoebius \u03bcMinus\n\ninstance lbTo\u03bcMinus : CoeFun LowerBoundSieve fun _ => \u2115 \u2192 \u211d where coe lb := lb.\u03bcMinus\n\nend UpperBoundSieve\n\nsection SieveLemmas\n\ntheorem prodPrimes_ne_zero : P \u2260 0 :=\n  Squarefree.ne_zero s.prodPrimes_squarefree\n\ntheorem squarefree_of_dvd_prodPrimes {d : \u2115} (hd : d \u2223 P) : Squarefree d :=\n  Squarefree.squarefree_of_dvd hd s.prodPrimes_squarefree\n\n", "theoremStatement": "theorem squarefree_of_mem_divisors_prodPrimes {d : \u2115} (hd : d \u2208 divisors P) : Squarefree d ", "theoremName": "Sieve.squarefree_of_mem_divisors_prodPrimes", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/Basic.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "jsonFile": 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"Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", 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"Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", 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"Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp only [Nat.mem_divisors, Ne.def] at hd\n  exact Squarefree.squarefree_of_dvd hd.left s.prodPrimes_squarefree", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 119}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Basic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.Wiener\n\nopen Asymptotics Complex ComplexConjugate Topology Filter Real MeasureTheory Set\n\nopen scoped Interval\n\n-- TODO: why do we need to bump this?\ninstance : MeasurableDiv\u2082 \u211d := by\n  haveI (G : Type) [DivInvMonoid G] [MeasurableSpace G] [MeasurableInv G] [MeasurableMul\u2082 G] :\n    MeasurableDiv\u2082 G := inferInstance\n  exact this \u211d\n\n/-%%\nIn this section, we prove the Perron formula, which plays a key role in our proof of Mellin inversion.\n%%-/\n\n/-%%\nThe following is preparatory material used in the proof of the Perron formula, see Lemma \\ref{formulaLtOne}.\n%%-/\n\n/-%\nTODO: move to general section.\n\\begin{lemma}[zeroTendstoDiff]\\label{zeroTendstoDiff}\\lean{zeroTendstoDiff}\\leanok\nIf the limit of $0$ is $L\u2081 - L\u2082$, then $L\u2081 = L\u2082$.\n\\end{lemma}\n%-/\nlemma zeroTendstoDiff (L\u2081 L\u2082 : \u2102) (f : \u211d \u2192 \u2102) (h : \u2200\u1da0 T in atTop,  f T = 0)\n    (h' : Tendsto f atTop (\ud835\udcdd (L\u2082 - L\u2081))) : L\u2081 = L\u2082 := by\n  rw [\u2190 zero_add L\u2081, \u2190 @eq_sub_iff_add_eq]\n  exact tendsto_nhds_unique (EventuallyEq.tendsto h) h'\n/-%\n\\begin{proof}\\leanok\nObvious.\n\\end{proof}\n%-/\n\n/-%\nTODO: Move this to general section.\n\\begin{lemma}[RectangleIntegral_tendsTo_VerticalIntegral]\\label{RectangleIntegral_tendsTo_VerticalIntegral}\\lean{RectangleIntegral_tendsTo_VerticalIntegral}\\leanok\n\\uses{RectangleIntegral}\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\lim_{T\\to\\infty}\\int_{\\sigma-iT}^{\\sigma'+iT}f(s)ds =\n\\int_{(\\sigma')}f(s)ds - \\int_{(\\sigma)}f(s)ds.$$\n\\end{lemma}\n%-/\n", "theoremStatement": "lemma RectangleIntegral_tendsTo_VerticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (T : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) atTop\n      (\ud835\udcdd (VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3)) ", "theoremName": 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"Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Integral.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n/-%\n\\begin{proof}\\leanok\nAlmost by definition.\n%-/\n  simp only [RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im, ofReal_im,\n    mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add, zero_sub,\n    add_im]\n  apply Tendsto.sub\n  \u00b7 rewrite [\u2190 zero_add (VerticalIntegral _ _), \u2190 zero_sub_zero]\n    apply Tendsto.add <| Tendsto.sub (hbot.comp tendsto_neg_atTop_atBot) htop\n    exact (intervalIntegral_tendsto_integral hright tendsto_neg_atTop_atBot tendsto_id).const_smul I\n  \u00b7 exact (intervalIntegral_tendsto_integral hleft tendsto_neg_atTop_atBot tendsto_id).const_smul I", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 621}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\n", "theoremStatement": "lemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) ", "theoremName": "dirichlet_test'", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", 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"Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 408}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\nlemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]\n\nlemma sum_eq_int_deriv_aux_lt {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_lt_kpOne : b < k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k := Int.floor_eq_iff.mpr \u27e8by linarith [ha.1, ha.2], by linarith\u27e9\n  simp only [flb_eq_k, gt_iff_lt, lt_add_iff_pos_right, zero_lt_one, Finset.Icc_eq_empty_of_lt,\n    Finset.sum_empty]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont]\n  have : Finset.Ioc k k = {} := by simp only [ge_iff_le, le_refl, Finset.Ioc_eq_empty_of_le]\n  simp only [this, Finset.sum_empty, one_div]; ring_nf\n\nlemma sum_eq_int_deriv_aux1 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  by_cases h : b = k + 1\n  \u00b7 exact sum_eq_int_deriv_aux_eq h \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact sum_eq_int_deriv_aux_lt ha (Ne.lt_of_le h b_le_kpOne) \u03c6Diff deriv\u03c6Cont\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv_aux]\\label{sum_eq_int_deriv_aux}\\lean{sum_eq_int_deriv_aux}\\leanok\n  Let $k \\le a < b\\le k+1$, with $k$ an integer, and let $\\phi$ be continuously differentiable on\n  $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma sum_eq_int_deriv_aux {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  have fl_a_eq_k : \u230aa\u230b = k := Int.floor_eq_iff.mpr \u27e8ha.1, by linarith [ha.2]\u27e9\n  convert sum_eq_int_deriv_aux1 ha b_le_kpOne \u03c6Diff deriv\u03c6Cont using 2\n  \u00b7 rw [fl_a_eq_k]\n  \u00b7 congr\n  \u00b7 apply intervalIntegral.integral_congr_ae\n    have : \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.volume, x \u2260 b := by\n      convert Countable.ae_not_mem (s := {b}) (by simp) (\u03bc := MeasureTheory.volume) using 1\n    filter_upwards [this]\n    intro x x_ne_b hx\n    rw [uIoc_of_le ha.2.le, mem_Ioc] at hx\n    congr\n    exact Int.floor_eq_iff.mpr \u27e8by linarith [ha.1], by have := Ne.lt_of_le x_ne_b hx.2; linarith\u27e9\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\n-- Thanks to Arend Mellendijk\n\nlemma interval_induction_aux_int (n : \u2115) : \u2200 (P : \u211d \u2192 \u211d \u2192 Prop)\n    (_ : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (_ : \u2200 (a : \u211d) (k : \u2124) (c : \u211d), a < k \u2192 k < c \u2192 P a k \u2192 P k c \u2192 P a c)\n    (a b : \u211d) (_ : a < b) (_ : n = \u230ab\u230b - \u230aa\u230b),\n    P a b := by\n  induction n using Nat.case_strong_induction_on with\n  | hz =>\n    intro P base _ a b hab hn\n    apply base a b \u230aa\u230b (Int.floor_le a) hab\n    rw [(by simp only [CharP.cast_eq_zero] at hn; linarith : \u230aa\u230b = \u230ab\u230b)]\n    exact (Int.lt_floor_add_one b).le\n  | hi n ih =>\n    intro P base step a b _ hn\n    have Pa : P a (\u230aa\u230b + 1) :=\n      base a (\u230aa\u230b + 1) \u230aa\u230b (Int.floor_le a) (Int.lt_floor_add_one a) (le_of_eq rfl)\n    by_cases b_le_flaP1 : b = \u230aa\u230b + 1\n    \u00b7 rwa [b_le_flaP1]\n    have flaP1_lt_b : \u230aa\u230b + 1 < b := by\n      simp only [Nat.cast_succ] at hn\n      have : (\u230aa\u230b : \u211d) + 1 \u2264 \u230ab\u230b := by exact_mod_cast (by linarith)\n      exact Ne.lt_of_le (id (Ne.symm b_le_flaP1)) (by linarith [Int.floor_le b] : \u230aa\u230b + 1 \u2264 b)\n    have Pfla_b : P (\u230aa\u230b + 1) b := by\n      apply ih n (le_of_eq rfl) P base step (\u230aa\u230b + 1) b flaP1_lt_b\n      simp only [Int.floor_add_one, Int.floor_intCast, Nat.cast_succ] at hn \u22a2\n      linarith\n    refine step a (\u230aa\u230b + 1) b ?_ (by exact_mod_cast flaP1_lt_b) (by exact_mod_cast Pa)\n      (by exact_mod_cast Pfla_b)\n    have := Int.lt_floor_add_one a\n    exact_mod_cast this\n\nlemma interval_induction (P : \u211d \u2192 \u211d \u2192 Prop)\n    (base : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (step : \u2200 (a : \u211d) (k : \u2124) (b : \u211d), a < k \u2192 k < b \u2192 P a k \u2192 P k b \u2192 P a b)\n    (a b : \u211d) (hab : a < b) : P a b := by\n  set n := \u230ab\u230b - \u230aa\u230b with hn\n  clear_value n\n  have : 0 \u2264 n := by simp only [hn, sub_nonneg, ge_iff_le, Int.floor_le_floor _ _ (hab.le)]\n  lift n to \u2115 using this\n  exact interval_induction_aux_int n P base step a b hab hn\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv]\\label{sum_eq_int_deriv}\\lean{sum_eq_int_deriv}\\leanok\n  Let $a < b$, and let $\\phi$ be continuously differentiable on $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\n/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {\u03b1 : Type*} [LinearOrder \u03b1] [LocallyFiniteOrder \u03b1]\n    {a b c: \u03b1} [DecidableEq \u03b1] (hb : b \u2208 Icc a c) : Ioc a b = Ioc a c \\ Ioc b c := by\n  ext x\n  simp only [mem_Ioc, mem_sdiff, not_and, not_le]\n  constructor\n  \u00b7 refine fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8\u27e8h\u2081, le_trans h\u2082 (mem_Icc.mp hb).2\u27e9, by contrapose! h\u2082; exact h\u2082.1\u27e9\n  \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8h\u2081.1, by contrapose! h\u2082; exact \u27e8h\u2082, h\u2081.2\u27e9\u27e9\n\n-- In Ya\u00ebl Dillies's API (https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Esum_add_adjacent_intervals/near/430127101)\nlemma Finset.sum_Ioc_add_sum_Ioc {a b c : \u2124} (f : \u2124 \u2192 \u2102) (hb : b \u2208 Icc a c):\n    (\u2211 n in Finset.Ioc a b, f n) + (\u2211 n in Finset.Ioc b c, f n) = \u2211 n in Finset.Ioc a c, f n := by\n  convert Finset.sum_sdiff (s\u2081 := Finset.Ioc b c) (s\u2082 := Finset.Ioc a c) ?_\n  \u00b7 exact Finset.Ioc_diff_Ioc hb\n  \u00b7 exact Finset.Ioc_subset_Ioc (mem_Icc.mp hb).1 (by rfl)\n\n", "theoremStatement": "lemma integrability_aux\u2080 {a b : \u211d} :\n    \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.Measure.restrict MeasureTheory.volume [[a, b]],\n      \u2016(\u230ax\u230b : \u2102)\u2016 \u2264 max \u2016a\u2016 \u2016b\u2016 + 1 ", "theoremName": "integrability_aux\u2080", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": 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"Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", 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"Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply (MeasureTheory.ae_restrict_iff' measurableSet_Icc).mpr\n  refine MeasureTheory.ae_of_all _ (fun x hx \u21a6 ?_)\n  simp only [inf_le_iff, le_sup_iff, mem_Icc] at hx\n  simp only [norm_int, Real.norm_eq_abs]\n  have : |x| \u2264 max |a| |b| := by\n    cases' hx.1 with x_ge_a x_ge_b <;> cases' hx.2 with x_le_a x_le_b\n    \u00b7 rw [(by linarith : x = a)]; apply le_max_left\n    \u00b7 apply abs_le_max_abs_abs x_ge_a x_le_b\n    \u00b7 rw [max_comm]; apply abs_le_max_abs_abs x_ge_b x_le_a\n    \u00b7 rw [(by linarith : x = b)]; apply le_max_right\n  cases' abs_cases x with hx hx\n  \u00b7 rw [_root_.abs_of_nonneg <| by exact_mod_cast Int.floor_nonneg.mpr hx.2]\n    apply le_trans (Int.floor_le x) <| le_trans (hx.1 \u25b8 this) (by simp)\n  \u00b7 rw [_root_.abs_of_nonpos <| by exact_mod_cast Int.floor_nonpos hx.2.le]\n    linarith [(Int.lt_floor_add_one x).le]", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 825}}
+{"srcContext": "import Mathlib.Analysis.Asymptotics.Theta\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\n\nnamespace Asymptotics\n\nvariable {\u03b1 \u03b2 E E' F : Type*} [Norm E] [SeminormedAddCommGroup E'] [Norm F] {f : \u03b1 \u2192 E} {g : \u03b1 \u2192 F}\n\n", "theoremStatement": "@[simp]\ntheorem isTheta_bot : f =\u0398[\u22a5] g ", "theoremName": "Asymptotics.isTheta_bot", "fileCreated": {"commit": "84fbebeb60a5e72e2d8a4e6b350a46af7023f681", "date": "2024-02-08"}, "theoremCreated": {"commit": "84fbebeb60a5e72e2d8a4e6b350a46af7023f681", "date": "2024-02-08"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Analysis/Asymptotics/Theta.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta.jsonl", "positionMetadata": {"lineInFile": 8, "tokenPositionInFile": 237, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, 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"Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics"]}, "proofMetadata": {"hasProof": true, "proof": ":= by simp [IsTheta]", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 20}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\n", "theoremStatement": "lemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x ", "theoremName": "sum_eq_int_deriv_aux_eq", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "149077ef0ab50e9958c46be9a9c38b142497e913", "date": "2024-03-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": "PrimeNumberTheoremAnd.ZetaBounds.jsonl", "positionMetadata": {"lineInFile": 152, "tokenPositionInFile": 6623, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 282, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 636}}
+{"srcContext": "import Mathlib.Analysis.Distribution.SchwartzSpace\nimport Mathlib.MeasureTheory.Integral.IntegralEqImproper\nimport Mathlib.Topology.ContinuousFunction.Bounded\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.Inversion\n\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport PrimeNumberTheoremAnd.Sobolev\n\nopen FourierTransform Real Complex MeasureTheory Filter Topology BoundedContinuousFunction SchwartzMap VectorFourier BigOperators\n\nlocal instance {E : Type*} : Coe (E \u2192 \u211d) (E \u2192 \u2102) := \u27e8fun f n => f n\u27e9\n\nsection lemmas\n\n@[simp]\ntheorem nnnorm_eq_of_mem_circle (z : circle) : \u2016z.val\u2016\u208a = 1 := NNReal.coe_eq_one.mp (by simp)\n\n@[simp]\ntheorem nnnorm_circle_smul (z : circle) (s : \u2102) : \u2016z \u2022 s\u2016\u208a = \u2016s\u2016\u208a := by\n  simp [show z \u2022 s = z.val * s from rfl]\n\nnoncomputable def e (u : \u211d) : \u211d \u2192\u1d47 \u2102 where\n  toFun v := \ud835\udc1e (-v * u)\n  map_bounded' := \u27e82, fun x y => (dist_le_norm_add_norm _ _).trans (by simp [one_add_one_eq_two])\u27e9\n\n@[simp] lemma e_apply (u : \u211d) (v : \u211d) : e u v = \ud835\udc1e (-v * u) := rfl\n\ntheorem hasDerivAt_e {u x : \u211d} : HasDerivAt (e u) (-2 * \u03c0 * u * I * e u x) x := by\n  have l2 : HasDerivAt (fun v => -v * u) (-u) x := by simpa only [neg_mul_comm] using hasDerivAt_mul_const (-u)\n  convert (hasDerivAt_fourierChar (-x * u)).scomp x l2 using 1\n  simp ; ring\n\nlemma fourierIntegral_deriv_aux2 (e : \u211d \u2192\u1d47 \u2102) {f : \u211d \u2192 \u2102} (hf : Integrable f) : Integrable (\u21d1e * f) :=\n  hf.bdd_mul e.continuous.aestronglyMeasurable \u27e8_, e.norm_coe_le_norm\u27e9\n\n@[simp] lemma F_neg {f : \u211d \u2192 \u2102} {u : \u211d} : \ud835\udcd5 (fun x => -f x) u = - \ud835\udcd5 f u := by\n  simp [fourierIntegral_eq, integral_neg]\n\n@[simp] lemma F_add {f g : \u211d \u2192 \u2102} (hf : Integrable f) (hg : Integrable g) (x : \u211d) :\n    \ud835\udcd5 (fun x => f x + g x) x = \ud835\udcd5 f x + \ud835\udcd5 g x :=\n  congr_fun (fourierIntegral_add continuous_fourierChar (by exact continuous_mul) hf hg).symm x\n\n@[simp] lemma F_sub {f g : \u211d \u2192 \u2102} (hf : Integrable f) (hg : Integrable g) (x : \u211d) :\n    \ud835\udcd5 (fun x => f x - g x) x = \ud835\udcd5 f x - \ud835\udcd5 g x := by\n  simp_rw [sub_eq_add_neg] ; rw [F_add] ; simp ; exact hf ; exact hg.neg\n\n@[simp] lemma F_mul {f : \u211d \u2192 \u2102} {c : \u2102} {u : \u211d} : \ud835\udcd5 (fun x => c * f x) u = c * \ud835\udcd5 f u := by\n  simp [fourierIntegral_real_eq, \u2190 integral_mul_left] ; congr ; ext\n  simp [Real.fourierChar, expMapCircle] ; ring\n\nend lemmas\n\ntheorem fourierIntegral_self_add_deriv_deriv (f : W21) (u : \u211d) :\n    (1 + u ^ 2) * \ud835\udcd5 f u = \ud835\udcd5 (fun u => f u - (1 / (4 * \u03c0 ^ 2)) * deriv^[2] f u) u := by\n  have l1 : Integrable (fun x => (((\u03c0 : \u2102) ^ 2)\u207b\u00b9 * 4\u207b\u00b9) * deriv (deriv f) x) := by\n    apply Integrable.const_mul ; simpa [iteratedDeriv_succ] using f.integrable le_rfl\n  have l4 : Differentiable \u211d f := f.differentiable\n  have l5 : Differentiable \u211d (deriv f) := f.deriv.differentiable\n  simp [f.hf, l1, add_mul, Real.fourierIntegral_deriv f.hf' l5 f.hf'', Real.fourierIntegral_deriv f.hf l4 f.hf']\n  field_simp [pi_ne_zero] ; ring_nf ; simp\n\n@[simp] lemma deriv_ofReal : deriv ofReal' = fun _ => 1 := by\n  ext x ; exact ((hasDerivAt_id x).ofReal_comp).deriv\n\ntheorem bla (a : \u2102) (f : \u211d \u2192 \u2102) (n : \u2115) (hf : ContDiff \u211d n f) :\n    iteratedDeriv n (fun x \u21a6 a * x * f x) = fun x =>\n      a * x * iteratedDeriv n f x + n * a * iteratedDeriv (n - 1) f x := by\n\n  induction n with\n  | zero => simp\n  | succ n ih =>\n    have l0 : ContDiff \u211d n f := hf.of_succ\n    rw [iteratedDeriv_succ, ih l0] ; ext x\n    have l5 : ContDiff \u211d (\u2191(1 + n)) f := by convert hf using 1 ; simp ; ring\n    have l4 : DifferentiableAt \u211d (fun x \u21a6 iteratedDeriv n f x) x := by\n      have := ((l5.iterate_deriv' 1 n).differentiable le_rfl).differentiableAt (x := x)\n      simpa [iteratedDeriv_eq_iterate] using this\n    have l3 : DifferentiableAt \u211d (fun x \u21a6 a * \u2191x) x := by\n      apply DifferentiableAt.const_mul\n      exact (contDiff_ofReal.differentiable le_top).differentiableAt\n    have l1 : DifferentiableAt \u211d (fun x \u21a6 a * \u2191x * iteratedDeriv n f x) x := l3.mul l4\n    have l2 : DifferentiableAt \u211d (fun x \u21a6 \u2191n * a * iteratedDeriv (n - 1) f x) x := by\n      apply DifferentiableAt.const_mul\n      apply l5.differentiable_iteratedDeriv\n      norm_cast ; exact Nat.sub_le _ _ |>.trans_lt (by simp)\n    simp [deriv_add l1 l2, deriv_mul l3 l4, \u2190 iteratedDeriv_succ]\n    cases n <;> simp <;> ring\n\nnoncomputable def MS (a : \u2102) (f : \ud835\udce2(\u211d, \u2102)) : \ud835\udce2(\u211d, \u2102) where\n  toFun x := a * x * f x\n  smooth' := contDiff_const.mul contDiff_ofReal |>.mul f.smooth'\n  decay' k n := by\n    simp only [norm_iteratedFDeriv_eq_norm_iteratedDeriv]\n    simp_rw [bla a f n <| f.smooth'.of_le le_top]\n    obtain \u27e8C\u2081, hC\u2081\u27e9 := f.decay' (k + 1) n\n    obtain \u27e8C\u2082, hC\u2082\u27e9 := f.decay' k (n - 1)\n    use \u2016a\u2016 * C\u2081 + \u2016a\u2016 * n * C\u2082 ; intro x\n    have l2 := norm_add_le (a * x * iteratedDeriv n f x) (n * a * iteratedDeriv (n - 1) f x)\n    have l3 : 0 \u2264 \u2016x\u2016 ^ k := by positivity\n    apply (mul_le_mul_of_nonneg_left l2 l3).trans ; rw [mul_add] ; apply add_le_add\n    \u00b7 have : 0 \u2264 \u2016a\u2016 := by positivity\n      convert mul_le_mul_of_nonneg_left (hC\u2081 x) this using 1\n      simp [norm_iteratedFDeriv_eq_norm_iteratedDeriv, abs_eq_self.mpr pi_nonneg] ; ring_nf ; rfl\n    \u00b7 have : 0 \u2264 \u2016a\u2016 * n := by positivity\n      convert mul_le_mul_of_nonneg_left (hC\u2082 x) this using 1\n      simp [norm_iteratedFDeriv_eq_norm_iteratedDeriv, abs_eq_self.mpr pi_nonneg] ; ring_nf ; rfl\n\n@[simp] lemma MS_apply (a : \u2102) (f : \ud835\udce2(\u211d, \u2102)) (x : \u211d) : MS a f x = (a * x) \u2022 f x := rfl\n\nlemma MS_iterate (a : \u2102) (f : \ud835\udce2(\u211d, \u2102)) (n : \u2115) : (MS a)^[n] f = fun x : \u211d => (a * x) ^ n \u2022 f x := by\n  induction n generalizing f with\n  | zero => simp\n  | succ n ih => ext x ; simp [ih, pow_succ] ; ring\n\nlemma fourierIntegral_decay_aux (f : \u211d \u2192 \u2102) (k : \u2115) (h1 : ContDiff \u211d k f)\n    (h2 : \u2200 n \u2264 k, Integrable (iteratedDeriv n f)) (x : \u211d) :\n    \u2016(2 * \u03c0 * I * x) ^ k \u2022 \ud835\udcd5 f x\u2016 \u2264 (\u222b y : \u211d, \u2016iteratedDeriv k f y\u2016) := by\n  have l2 (x : \u211d) : (2 * \u03c0 * I * x) ^ k \u2022 \ud835\udcd5 f x = \ud835\udcd5 (iteratedDeriv k f) x := by\n    simp [Real.fourierIntegral_iteratedDeriv h1 (fun n hn => h2 n <| Nat.cast_le.mp hn) le_rfl]\n  simpa only [l2] using Fourier.norm_fourierIntegral_le_integral_norm ..\n\nlemma iteratedDeriv_schwartz (f : \ud835\udce2(\u211d, \u2102)) (n : \u2115) : iteratedDeriv n f = (SchwartzMap.derivCLM \u211d)^[n] f := by\n  induction n with\n  | zero => rfl\n  | succ n ih => rw [iteratedDeriv_succ, ih, Function.iterate_succ'] ; rfl\n\ntheorem fourierIntegral_decay (f : \ud835\udce2(\u211d, \u2102)) (k : \u2115) : \u2203 C, \u2200 (x : \u211d), \u2016x\u2016 ^ k * \u2016\ud835\udcd5 f x\u2016 \u2264 C := by\n  convert_to \u2203 C, \u2200 x : \u211d, \u2016x ^ k * \ud835\udcd5 f x\u2016 \u2264 C ; \u00b7 simp\n  convert_to \u2203 C, \u2200 x : \u211d, \u2016(2 * \u03c0 * I * x) ^ k * \ud835\udcd5 f x\u2016 / (2 * \u03c0) ^ k \u2264 C using 4\n  \u00b7 field_simp [mul_pow, abs_eq_self.mpr pi_nonneg] ; ring\n  convert_to \u2203 C, \u2200 x : \u211d, \u2016(2 * \u03c0 * I * x) ^ k \u2022 \ud835\udcd5 f x\u2016 / (2 * \u03c0) ^ k \u2264 C\n  use (\u222b (y : \u211d), \u2016iteratedDeriv k (\u21d1f) y\u2016) / (2 * \u03c0) ^ k ; intro x\n  have l1 : \u2200 n \u2264 k, Integrable (iteratedDeriv n f) volume := by\n    simp_rw [iteratedDeriv_schwartz] ; simp [SchwartzMap.integrable]\n  have := fourierIntegral_decay_aux f k (f.smooth'.of_le le_top) l1 x\n  apply div_le_div_of_nonneg_right this (by positivity)\n\nnoncomputable def FS (f : \ud835\udce2(\u211d, \u2102)) : \ud835\udce2(\u211d, \u2102) where\n  toFun := \ud835\udcd5 f\n  smooth' := by\n    rw [contDiff_top] ; intro n\n    apply Real.contDiff_fourierIntegral ; intro k _\n    apply SchwartzMap.integrable_pow_mul\n  decay' := by\n    simp only [norm_iteratedFDeriv_eq_norm_iteratedDeriv]\n    intro k n\n    have l1 (k : \u2115) (_ : k \u2264 (n : \u2115\u221e)) : Integrable (fun x \u21a6 x ^ k \u2022 f x) volume := by\n      convert_to Integrable ((MS 1)^[k] f) ; \u00b7 simp [MS_iterate]\n      apply SchwartzMap.integrable\n    simp_rw [@Real.iteratedDeriv_fourierIntegral \u2102 _ _ f n n l1 le_rfl]\n    convert_to \u2203 C, \u2200 (x : \u211d), \u2016x\u2016 ^ k * \u2016\ud835\udcd5 ((MS (-2 * \u03c0 * I))^[n] f) x\u2016 \u2264 C ; \u00b7 simp [MS_iterate]\n    apply fourierIntegral_decay\n\n@[simp] lemma FS_apply (f : \ud835\udce2(\u211d, \u2102)) (x : \u211d) : FS f x = \ud835\udcd5 f x := rfl\n\n@[simp] lemma FS_toFun (f : \ud835\udce2(\u211d, \u2102)) : \u21d1(FS f) = \ud835\udcd5 f := rfl\n\n", "theoremStatement": "@[simp] lemma schwarz_reduce (f : \u211d \u2192 \u2102) h1 h2 x : SchwartzMap.mk f h1 h2 x = f x ", "theoremName": "schwarz_reduce", "fileCreated": {"commit": "00a2f4cf69f1409c26fcd9d398e985f4b733128e", "date": "2024-03-20"}, "theoremCreated": {"commit": "53de16ec5814caedef08dce9b306b92924a08717", "date": "2024-04-08"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Fourier.lean", "module": "PrimeNumberTheoremAnd.Fourier", "jsonFile": "PrimeNumberTheoremAnd.Fourier.jsonl", "positionMetadata": {"lineInFile": 159, "tokenPositionInFile": 7693, 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"Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\n", "theoremStatement": "lemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) ", "theoremName": "Filter.BigO_zero_atZero_of_support_in_Icc", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "009d47f1de1b230f1dd87c588e239f8867f6c2e3", "date": "2024-03-30"}, "file": 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"Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", 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"PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 254}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\nlemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self\n\nlemma bound_sum_log0 {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let f0 i := if i = 0 then 0 else f i\n  have l1 : chebyWith C f0 := by\n    intro n ; refine Finset.sum_le_sum (fun i _ => ?_) |>.trans (hf n)\n    by_cases hi : i = 0 <;> simp [hi, f0]\n  have l2 i : \u2016f i\u2016 / i = \u2016f0 i\u2016 / i := by by_cases hi : i = 0 <;> simp [hi, f0]\n  simp_rw [l2] ; apply bound_sum_log rfl l1 hx\n\nlemma bound_sum_log' {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + 2 * \u03c0 ^ 2) := by\n  simpa only [hh_integral'] using bound_sum_log0 hf hx\n\nlemma summable_fourier (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  exact Summable.of_nonneg_of_le (fun _ => norm_nonneg _) l6 (by simpa using l5.const_smul (W21.norm \u03c8))\n\nlemma bound_I1 (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264\n    W21.norm \u03c8 \u2022 \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 := by\n\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  have l1 : Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n    exact summable_fourier x hx \u03c8 hcheby\n  apply (norm_tsum_le_tsum_norm l1).trans\n  simpa only [\u2190 tsum_const_smul _ l5] using tsum_mono l1 (by simpa using l5.const_smul (W21.norm \u03c8)) l6\n\nlemma bound_I1' {C : \u211d} (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21) (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264 W21.norm \u03c8 * C * (1 + 2 * \u03c0 ^ 2) := by\n\n  apply bound_I1 x (by linarith) \u03c8 \u27e8_, hcheby\u27e9 |>.trans\n  rw [smul_eq_mul, mul_assoc]\n  apply mul_le_mul le_rfl (bound_sum_log' hcheby hx) ?_ W21.norm_nonneg\n  apply tsum_nonneg (fun i => by positivity)\n\nlemma bound_I2 (x : \u211d) (\u03c8 : W21) :\n    \u2016\u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (2 * \u03c0 ^ 2) := by\n\n  have key a : \u2016\ud835\udcd5 \u03c8 (a / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := decay_bounds_key \u03c8 _\n  have twopi : 0 \u2264 2 * \u03c0 := by simp [pi_nonneg]\n  have l3 : Integrable (fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.comp_div (by norm_num [pi_ne_zero])\n  have l2 : IntegrableOn (fun i \u21a6 W21.norm \u03c8 * (1 + (i / (2 * \u03c0)) ^ 2)\u207b\u00b9) (Ici (-Real.log x)) := by\n    exact (l3.const_mul _).integrableOn\n  have l1 : IntegrableOn (fun i \u21a6 \u2016\ud835\udcd5 \u03c8 (i / (2 * \u03c0))\u2016) (Ici (-Real.log x)) := by\n    refine ((l3.const_mul (W21.norm \u03c8)).mono' ?_ ?_).integrableOn\n    \u00b7 apply Continuous.aestronglyMeasurable ; continuity\n    \u00b7 simp only [norm_norm, key] ; simp\n  have l5 : 0 \u2264\u1d50[volume] fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := by apply eventually_of_forall ; intro x ; positivity\n  refine (norm_integral_le_integral_norm _).trans <| (set_integral_mono l1 l2 key).trans ?_\n  rw [integral_mul_left] ; gcongr ; apply W21.norm_nonneg\n  refine (set_integral_le_integral l3 l5).trans ?_\n  rw [Measure.integral_comp_div (fun x => (1 + x ^ 2)\u207b\u00b9) (2 * \u03c0)]\n  simp [abs_eq_self.mpr twopi] ; ring_nf ; rfl\n\nlemma bound_main {C : \u211d} (A : \u2102) (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21)\n    (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264\n      W21.norm \u03c8 * (C * (1 + 2 * \u03c0 ^ 2) + \u2016A\u2016 * (2 * \u03c0 ^ 2)) := by\n\n  have l1 := bound_I1' x hx \u03c8 hcheby\n  have l2 := mul_le_mul (le_refl \u2016A\u2016) (bound_I2 x \u03c8) (by positivity) (by positivity)\n  apply norm_sub_le _ _ |>.trans ; rw [norm_mul]\n  convert _root_.add_le_add l1 l2 using 1 ; ring\n\n/-%%\n\\begin{lemma}[Limiting identity for Schwartz functions]\\label{schwarz-id}\\lean{limiting_cor_schwartz}\\leanok  The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.\n\\end{lemma}\n%%-/\n\nlemma limiting_cor_W21 (\u03c8 : W21) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) := by\n\n  -- Shorter notation for clarity\n  let S1 x (\u03c8 : \u211d \u2192 \u2102) := \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191n / x))\n  let S2 x (\u03c8 : \u211d \u2192 \u2102) := \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\n  let S x \u03c8 := S1 x \u03c8 - S2 x \u03c8 ; change Tendsto (fun x \u21a6 S x \u03c8) atTop (\ud835\udcdd 0)\n\n  -- Build the truncation\n  obtain g := exists_trunc\n  let \u03a8 R := g.scale R * \u03c8\n  have key R : Tendsto (fun x \u21a6 S x (\u03a8 R)) atTop (\ud835\udcdd 0) := limiting_cor (\u03a8 R) hf hcheby hG hG'\n\n  -- Choose the truncation radius\n  obtain \u27e8C, hcheby\u27e9 := hcheby\n  have hC : 0 \u2264 C := by\n    have : \u2016f 0\u2016 \u2264 C := by simpa [cumsum] using hcheby 1\n    have : 0 \u2264 \u2016f 0\u2016 := by positivity\n    linarith\n  have key2 : Tendsto (fun R \u21a6 W21.norm (\u03c8 - \u03a8 R)) atTop (\ud835\udcdd 0) := W21_approximation \u03c8 g\n  simp_rw [Metric.tendsto_nhds] at key key2 \u22a2 ; intro \u03b5 h\u03b5\n  let M := C * (1 + 2 * \u03c0 ^ 2) + \u2016(A : \u2102)\u2016 * (2 * \u03c0 ^ 2)\n  obtain \u27e8R, hR\u03c8\u27e9 := (key2 ((\u03b5 / 2) / (1 + M)) (by positivity)).exists\n  simp only [dist_zero_right, Real.norm_eq_abs, abs_eq_self.mpr W21.norm_nonneg] at hR\u03c8 key\n\n  -- Apply the compact support case\n  filter_upwards [eventually_ge_atTop 1, key R (\u03b5 / 2) (by positivity)] with x hx key\n\n  -- Control the tail term\n  have key3 : \u2016S x (\u03c8 - \u03a8 R)\u2016 < \u03b5 / 2 := by\n    have : \u2016S x _\u2016 \u2264 _ * M := @bound_main f C A x hx (\u03c8 - \u03a8 R) hcheby\n    apply this.trans_lt\n    apply (mul_le_mul (d := 1 + M) le_rfl (by simp) (by positivity) W21.norm_nonneg).trans_lt\n    have : 0 < 1 + M := by positivity\n    convert (mul_lt_mul_right this).mpr hR\u03c8 using 1 ; field_simp ; ring\n\n  -- Conclude the proof\n  have S1_sub_1 x : \ud835\udcd5 (\u21d1\u03c8 - \u21d1(\u03a8 R)) x = \ud835\udcd5 \u03c8 x - \ud835\udcd5 (\u03a8 R) x := by\n    have l1 : AEStronglyMeasurable (fun x_1 : \u211d \u21a6 cexp (-(2 * \u2191\u03c0 * (\u2191x_1 * \u2191x) * I))) volume := by\n      refine (Continuous.mul ?_ continuous_const).neg.cexp.aestronglyMeasurable\n      apply continuous_const.mul <| contDiff_ofReal.continuous.mul continuous_const\n    simp [Real.fourierIntegral_eq', mul_sub] ; apply integral_sub\n    \u00b7 apply \u03c8.hf.bdd_mul l1 ; use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n    \u00b7 apply (\u03a8 R : W21) |>.hf |>.bdd_mul l1\n      use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n\n  have S1_sub : S1 x (\u03c8 - \u03a8 R) = S1 x \u03c8 - S1 x (\u03a8 R) := by\n    simp [S1, S1_sub_1, mul_sub] ; apply tsum_sub\n    \u00b7 have := summable_fourier x (by positivity) \u03c8 \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n    \u00b7 have := summable_fourier x (by positivity) (\u03a8 R) \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n\n  have S2_sub : S2 x (\u03c8 - \u03a8 R) = S2 x \u03c8 - S2 x (\u03a8 R) := by\n    simp [S2, S1_sub_1] ; rw [integral_sub] ; ring\n    \u00b7 exact \u03c8.integrable_fourier (by positivity) |>.restrict\n    \u00b7 exact (\u03a8 R : W21).integrable_fourier (by positivity) |>.restrict\n\n  have S_sub : S x (\u03c8 - \u03a8 R) = S x \u03c8 - S x (\u03a8 R) := by simp [S, S1_sub, S2_sub] ; ring\n  simpa [S_sub, \u03a8] using norm_add_le _ _ |>.trans_lt (_root_.add_lt_add key3 key)\n\nlemma limiting_cor_schwartz (\u03c8 : \ud835\udce2(\u211d, \u2102)) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) :=\n  limiting_cor_W21 \u03c8 hf hcheby hG hG'\n\n/-%%\n\\begin{proof}\n\\uses{limiting-cor, smooth-ury}\\leanok\nFor any $R>1$, one can use a smooth cutoff function (provided by Lemma \\ref{smooth-ury} to write $\\psi = \\psi_{\\leq R} + \\psi_{>R}$, where $\\psi_{\\leq R}$ is $C^2$ (in fact smooth) and compactly supported (on $[-R,R]$), and $\\psi_{>R}$ obeys bounds of the form\n$$ |\\psi_{>R}(t)|, |\\psi''_{>R}(t)| \\ll R^{-1} / (1 + |t|^2) $$\nwhere the implied constants depend on $\\psi$.  By Lemma \\ref{decay} we then have\n$$ \\hat \\psi_{>R}(u) \\ll R^{-1} / (1+|u|^2).$$\nUsing this and \\eqref{cheby} one can show that\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{>R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ), A \\int_{-\\infty}^\\infty \\hat \\psi_{>R} (\\frac{u}{2\\pi})\\ du \\ll R^{-1} $$\n(with implied constants also depending on $A$), while from Lemma \\ref{limiting-cor} one has\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{\\leq R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi_{\\leq R} (\\frac{u}{2\\pi})\\ du + o(1).$$\nCombining the two estimates and letting $R$ be large, we obtain the claim.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Bijectivity of Fourier transform]\\label{bij}\\lean{fourier_surjection_on_schwartz}\\leanok  The Fourier transform is a bijection on the Schwartz class.\n\\end{lemma}\n%%-/\n\n-- just the surjectivity is stated here, as this is all that is needed for the current application, but perhaps one should state and prove bijectivity instead\n\nlemma fourier_surjection_on_schwartz (f : \ud835\udce2(\u211d, \u2102)) : \u2203 g : \ud835\udce2(\u211d, \u2102), \ud835\udcd5 g = f := by\n  use FS (FS (FS f)) ; ext x ; nth_rewrite 2 [\u2190 FS4 f] ; simp\n\n/-%%\n\\begin{proof}\n  \\leanok\n This is a standard result in Fourier analysis.\nIt can be proved here by appealing to Mellin inversion, Theorem \\ref{MellinInversion}.\nIn particular, given $f$ in the Schwartz class, let $F : \\R_+ \\to \\C : x \\mapsto f(\\log x)$ be a function in the ``Mellin space''; then the Mellin transform of $F$ on the imaginary axis $s=it$ is the Fourier transform of $f$.  The Mellin inversion theorem gives Fourier inversion.\n\\end{proof}\n%%-/\n\ndef toSchwartz (f : \u211d \u2192 \u2102) (h1 : ContDiff \u211d \u22a4 f) (h2 : HasCompactSupport f) : \ud835\udce2(\u211d, \u2102) where\n  toFun := f\n  smooth' := h1\n  decay' k n := by\n    have l1 : Continuous (fun x => \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := by\n      have : ContDiff \u211d \u22a4 (iteratedFDeriv \u211d n f) := h1.iteratedFDeriv_right le_top\n      exact Continuous.mul (by continuity) this.continuous.norm\n    have l2 : HasCompactSupport (fun x \u21a6 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := (h2.iteratedFDeriv _).norm.mul_left\n    simpa using l1.bounded_above_of_compact_support l2\n\n@[simp] lemma toSchwartz_apply (f : \u211d \u2192 \u2102) {h1 h2 x} : SchwartzMap.mk f h1 h2 x = f x := rfl\n\nlemma comp_exp_support0 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in \ud835\udcdd 0, \u03a8 x = 0 :=\n  not_mem_tsupport_iff_eventuallyEq.mp (fun h => lt_irrefl 0 <| mem_Ioi.mp (hplus h))\n\nlemma comp_exp_support1 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in atBot, \u03a8 (exp x) = 0 :=\n  Real.tendsto_exp_atBot <| comp_exp_support0 hplus\n\nlemma comp_exp_support2 {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) :\n    \u2200\u1da0 (x : \u211d) in atTop, (\u03a8 \u2218 rexp) x = 0 := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop] at hsupp\n  exact Real.tendsto_exp_atTop hsupp.2\n\ntheorem comp_exp_support {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    HasCompactSupport (\u03a8 \u2218 rexp) := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop]\n  exact \u27e8comp_exp_support1 hplus, comp_exp_support2 hsupp\u27e9\n\n", "theoremStatement": "lemma wiener_ikehara_smooth_aux (l0 : Continuous \u03a8) (hsupp : HasCompactSupport \u03a8)\n    (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) (x : \u211d) (hx : 0 < x) :\n    \u222b (u : \u211d) in Ioi (-Real.log x), \u2191(rexp u) * \u03a8 (rexp u) = \u222b (y : \u211d) in Ioi (1 / x), \u03a8 y ", "theoremName": "wiener_ikehara_smooth_aux", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "b9c184ee67db654d130697074bfb181235c3288c", "date": "2024-04-02"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1641, "tokenPositionInFile": 81508, "theoremPositionInFile": 136}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, 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"Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  have l1 : ContinuousOn rexp (Ici (-Real.log x)) := by fun_prop\n  have l2 : Tendsto rexp atTop atTop := Real.tendsto_exp_atTop\n  have l3 t (_ : t \u2208 Ioi (-log x)) : HasDerivWithinAt rexp (rexp t) (Ioi t) t :=\n    (Real.hasDerivAt_exp t).hasDerivWithinAt\n  have l4 : ContinuousOn \u03a8 (rexp '' Ioi (-Real.log x)) := by fun_prop\n  have l5 : IntegrableOn \u03a8 (rexp '' Ici (-Real.log x)) volume :=\n    (l0.integrable_of_hasCompactSupport hsupp).integrableOn\n  have l6 : IntegrableOn (fun x \u21a6 rexp x \u2022 (\u03a8 \u2218 rexp) x) (Ici (-Real.log x)) volume := by\n    refine (Continuous.integrable_of_hasCompactSupport (by continuity) ?_).integrableOn\n    change HasCompactSupport (rexp \u2022 (\u03a8 \u2218 rexp))\n    exact (comp_exp_support hsupp hplus).smul_left\n  have := MeasureTheory.integral_comp_smul_deriv_Ioi l1 l2 l3 l4 l5 l6\n  simpa [Real.exp_neg, Real.exp_log hx] using this", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 855}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]\n\nlemma Filter.BigO_zero_atTop_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[atTop] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioi_mem_atTop b] with c hc; replace hc := mem_Ioi.mp hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h]\n\n-- steal coerction lemmas from EulerProducts.Auxiliary because of build issues, and add new ones\nnamespace Complex\n-- see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Differentiability.20of.20the.20natural.20map.20.E2.84.9D.20.E2.86.92.20.E2.84.82/near/418095234\n\nlemma hasDerivAt_ofReal (x : \u211d) : HasDerivAt ofReal' 1 x :=\n  HasDerivAt.ofReal_comp <| hasDerivAt_id x\n\nlemma deriv_ofReal (x : \u211d) : deriv ofReal' x = 1 :=\n  (hasDerivAt_ofReal x).deriv\n\nlemma differentiableAt_ofReal (x : \u211d) : DifferentiableAt \u211d ofReal' x :=\n  (hasDerivAt_ofReal x).differentiableAt\n\nlemma differentiable_ofReal : Differentiable \u211d ofReal' :=\n  ofRealCLM.differentiable\n\nend Complex\n\nlemma DifferentiableAt.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    DifferentiableAt \u211d (fun x : \u211d \u21a6 e x) z :=\n  hf.hasDerivAt.comp_ofReal.differentiableAt\n\nlemma Differentiable.comp_ofReal {e : \u2102 \u2192 \u2102} (h : Differentiable \u2102 e) :\n    Differentiable \u211d (fun x : \u211d \u21a6 e x) :=\n  fun _ \u21a6 h.differentiableAt.comp_ofReal\n\nlemma DifferentiableAt.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} (hf : DifferentiableAt \u211d f z) :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z :=\n  hf.hasDerivAt.ofReal_comp.differentiableAt\n\nlemma Differentiable.ofReal_comp {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) :=\n  fun _ \u21a6 hf.differentiableAt.ofReal_comp\n\nopen Complex ContinuousLinearMap in\nlemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} {u : \u2102}\n    (hf : HasDerivAt (fun y \u21a6 (f y : \u2102)) u z) :\n    \u2203 u' : \u211d, u = u' \u2227 HasDerivAt f u' z := by\n  lift u to \u211d\n  \u00b7 have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv\n    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H\n    rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H\n  refine \u27e8u, rfl, ?_\u27e9\n  convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt\n  rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]\n\nlemma DifferentiableAt.ofReal_comp_iff {z : \u211d} {f : \u211d \u2192 \u211d} :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z \u2194 DifferentiableAt \u211d f z := by\n  refine \u27e8fun H \u21a6 ?_, ofReal_comp\u27e9\n  obtain \u27e8u, _, hu\u2082\u27e9 := H.hasDerivAt.of_hasDerivAt_ofReal_comp\n  exact HasDerivAt.differentiableAt hu\u2082\n\nlemma Differentiable.ofReal_comp_iff {f : \u211d \u2192 \u211d} :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) \u2194 Differentiable \u211d f :=\n  forall_congr' fun _ \u21a6 DifferentiableAt.ofReal_comp_iff\n\nlemma deriv.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} :\n    deriv (fun (y : \u211d) \u21a6 (f y : \u2102)) z = deriv f z := by\n  by_cases hf : DifferentiableAt \u211d f z\n  \u00b7 exact hf.hasDerivAt.ofReal_comp.deriv\n  \u00b7 have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf\n    rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt <| hf',\n      Complex.ofReal_zero]\n\nlemma deriv.ofReal_comp' {f : \u211d \u2192 \u211d} :\n    deriv (fun x : \u211d \u21a6 (f x : \u2102)) = (fun x \u21a6 ((deriv f) x : \u2102)) :=\n  funext fun _ \u21a6 deriv.ofReal_comp\n\nlemma deriv.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    deriv (fun x : \u211d \u21a6 e x) z = deriv e z :=\n  hf.hasDerivAt.comp_ofReal.deriv\n\nlemma deriv.comp_ofReal' {e : \u2102 \u2192 \u2102} (hf : Differentiable \u2102 e) :\n    deriv (fun x : \u211d \u21a6 e x) = fun (x : \u211d) \u21a6 deriv e x :=\n  funext fun _ \u21a6 deriv.comp_ofReal (hf.differentiableAt)\n\n/-%%\n\\begin{lemma}[PartialIntegration]\\label{PartialIntegration}\\lean{PartialIntegration}\\leanok\nLet $f, g$ be once differentiable functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ so that $fg'$\nand $f'g$ are both integrable, and $f*g (x)\\to 0$ as $x\\to 0^+,\\infty$.\nThen\n$$\n\\int_0^\\infty f(x)g'(x) dx = -\\int_0^\\infty f'(x)g(x)dx.\n$$\n\\end{lemma}\n%%-/\n/-- *Need differentiability, and decay at `0` and `\u221e`* -/\nlemma PartialIntegration (f g : \u211d \u2192 \u2102)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0))\n    (gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0))\n    (lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  simpa using integral_Ioi_mul_deriv_eq_deriv_mul\n    (fun x hx \u21a6 fDiff.hasDerivAt (Ioi_mem_nhds hx))\n    (fun x hx \u21a6 gDiff.hasDerivAt (Ioi_mem_nhds hx))\n    fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\nlemma PartialIntegration_of_support_in_Icc {a b : \u211d} (f g : \u211d \u2192 \u2102) (ha : 0 < a) (h : a \u2264 b)\n    (fSupp : f.support \u2286 Set.Icc a b)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fderivCont : ContinuousOn (deriv f) (Ioi 0))\n    (gderivCont : ContinuousOn (deriv g) (Ioi 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  have Icc_sub : Icc a b \u2286 Ioi 0 := (Icc_subset_Ioi_iff h).mpr ha\n  have fderivSupp := Function.support_deriv_subset_Icc fSupp\n  have fgSupp : (f * g).support \u2286 Icc a b := Function.support_mul_subset_of_subset fSupp\n  have fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fDiff.continuousOn.mono Icc_sub\n    \u00b7 exact gderivCont.mono Icc_sub\n  have gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fderivSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fderivCont.mono Icc_sub\n    \u00b7 exact gDiff.continuousOn.mono Icc_sub\n  have lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0) := TendstoAtZero_of_support_in_Icc (f * g) ha fgSupp\n  have lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0) := TendstoAtTop_of_support_in_Icc (f * g) fgSupp\n  apply PartialIntegration f g fDiff gDiff fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n\n/-%%\nIn this section, we define the Mellin transform (already in Mathlib, thanks to David Loeffler),\nprove its inversion formula, and\nderive a number of important properties of some special functions and bumpfunctions.\n\nDef: (Already in Mathlib)\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be the function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\n[Note: My preferred way to think about this is that we are integrating over the multiplicative\ngroup $\\mathbb{R}_{>0}$, multiplying by a (not necessarily unitary!) character $|\\cdot|^s$, and\nintegrating with respect to the invariant Haar measure $dx/x$. This is very useful in the kinds\nof calculations carried out below. But may be more difficult to formalize as things now stand. So\nwe might have clunkier calculations, which ``magically'' turn out just right - of course they're\nexplained by the aforementioned structure...]\n\n%%-/\n\n\n/-%%\n\\begin{definition}[MellinTransform]\\label{MellinTransform}\\lean{MellinTransform}\\leanok\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be\nthe function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\\end{definition}\n[Note: already exists in Mathlib, with some good API.]\n%%-/\nnoncomputable def MellinTransform (f : \u211d \u2192 \u2102) (s : \u2102) : \u2102 :=\n  \u222b x in Ioi 0, f x * x ^ (s - 1)\n\nlocal notation (name := mellintransform) \"\ud835\udcdc\" => MellinTransform\n/-%%\n\\begin{definition}[MellinInverseTransform]\\label{MellinInverseTransform}\n\\lean{MellinInverseTransform}\\leanok\nLet $F$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$. We define the Mellin inverse transform of\n$F$ to be the function $\\mathcal{M}^{-1}(F)$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}^{-1}(F)(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}F(s)x^{-s}ds,$$\nwhere $\\sigma$ is sufficiently large (say $\\sigma>2$).\n\\end{definition}\n%%-/\nnoncomputable def MellinInverseTransform (F : \u2102 \u2192 \u2102) (\u03c3 : \u211d) (x : \u211d) : \u2102 :=\n  VerticalIntegral' (fun s \u21a6 x ^ (-s) * F s) \u03c3\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_lt]\\label{PerronInverseMellin_lt}\\lean{PerronInverseMellin_lt}\n\\leanok\nLet $0 < t < x$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function\n$$F: s\\mapsto t^s/s(s+1)$$ is equal to\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds\n= 0.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_lt {t x : \u211d} (tpos : 0 < t) (t_lt_x : t < x) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 0 := by\n  dsimp [MellinInverseTransform, VerticalIntegral']\n  have xpos : 0 < x := lt_trans tpos t_lt_x\n  simp only [one_div, mul_inv_rev, inv_I, neg_mul, neg_eq_zero, mul_eq_zero, I_ne_zero,\n    inv_eq_zero, ofReal_eq_zero, pi_ne_zero, OfNat.ofNat_ne_zero, or_self, false_or]\n  convert Perron.formulaLtOne (div_pos tpos xpos) ((div_lt_one xpos).mpr t_lt_x) \u03c3pos using 2\n  ext1 s\n  convert Perron.f_mul_eq_f tpos xpos s using 1\n  ring\n/-%%\n\\begin{proof}\\leanok\n\\uses{Perron.formulaLtOne}\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_gt]\\label{PerronInverseMellin_gt}\\lean{PerronInverseMellin_gt}\n\\leanok\nLet $0 < x < t$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function is equal\nto\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds = 1 - x / t.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_gt {t x : \u211d} (xpos : 0 < x) (x_lt_t : x < t) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 1 - x / t := by\n  dsimp [MellinInverseTransform]\n  have tpos : 0 < t := by linarith\n  have txinv_gtOne : 1 < t / x := (one_lt_div xpos).mpr x_lt_t\n  rw [\u2190 smul_eq_mul]\n  convert Perron.formulaGtOne txinv_gtOne \u03c3pos using 2\n  \u00b7 congr\n    ext1 s\n    convert Perron.f_mul_eq_f tpos xpos s using 1\n    ring\n  \u00b7 field_simp\n/-%%\n\\begin{proof}\n\\uses{Perron.formulaGtOne}\\leanok\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-% ** Wrong delimiters on purpose **\nUnnecessary lemma:\n%\\begin{lemma}[MellinInversion_aux1]\\label{MellinInversion_aux1}\\lean{MellinInversion_aux1}\\leanok\nLet $f$ be differentiable on $(0,\\infty)$, and assume that $f(x)x^s\\to 0$ as $x\\to 0$, and that\n$f(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f(x)x^s\\frac{dx}{x} = \\frac{1}{s}\\int_0^\\infty f'(x)x^{s} dx.\n$$\n%\\end{lemma}\n%-/\nlemma MellinInversion_aux1 {f : \u211d \u2192 \u2102} {s : \u2102} (s_ne_zero : s \u2260 0)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * x ^ s / x = - \u222b x in Ioi 0, (deriv f x) * x ^ s / s := by\n  sorry\n\n/-% ** Wrong delimiters on purpose **\n\\begin{proof}\n\\uses{PartialIntegration}\nPartial integration.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux2]\\label{MellinInversion_aux2}\\lean{MellinInversion_aux2}\\leanok\nLet $f$ be twice differentiable on $(0,\\infty)$, and assume that $f'(x)x^s\\to 0$ as $x\\to 0$, and\nthat $f'(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f'(x)x^{s} dx = -\\int_0^\\infty f''(x)x^{s+1}\\frac{1}{(s+1)}dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux2 {f : \u211d \u2192 \u2102} (s : \u2102) (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (fDiff2 : DifferentiableOn \u211d (deriv f) (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 deriv f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 deriv f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, (deriv f x) * x ^ s =\n      -\u222b x in Ioi 0, (deriv (deriv f) x) * x ^ (s + 1) / (s + 1) := by\n  sorry\n/-%\n\\begin{proof}\n\\uses{PartialIntegration, MellinInversion_aux1}\nPartial integration. (Apply Lemma \\ref{MellinInversion_aux1} to the function $f'$ and $s+1$.)\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux3]%\\label{MellinInversion_aux3}\\lean{MellinInversion_aux3}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen the map  $(x,s) \\mapsto f(x)x^s/(s(s+1))$ is absolutely integrable on\n$(0,\\infty)\\times\\{\\Re s = \\sigma\\}$ for any $\\sigma>0$.\n\\end{lemma}\n%-/\nlemma MellinInversion_aux3 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    IntegrableOn (fun (\u27e8x, t\u27e9 : \u211d \u00d7 \u211d) \u21a6 f x * x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * ((\u03c3 + t * I) + 1)))\n      ((Ioi 0).prod (univ : Set \u211d)) := by\n  sorry\n/-%\n\\begin{proof}\nPut absolute values and estimate.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux4]%\\label{MellinInversion_aux4}\\lean{MellinInversion_aux4}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen we can interchange orders of integration\n$$\n\\int_{(\\sigma)}\\int_0^\\infty f(x)x^{s+1}\\frac{1}{s(s+1)}dx ds =\n\\int_0^\\infty\n\\int_{(\\sigma)}f(x)x^{s+1}\\frac{1}{s(s+1)}ds dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux4 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    VerticalIntegral (fun s \u21a6 \u222b x in Ioi 0, f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 =\n      \u222b x in Ioi 0, VerticalIntegral (fun s \u21a6 f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 := by\n  sorry -- `MeasureTheory.integral_prod` and `MeasureTheory.integral_swap` should be useful here\n/-%\n\\begin{proof}\n\\uses{MellinInversion_aux3}\nFubini-Tonelli.\n\\end{proof}\n%-/\n\nlemma MellinTransform_eq : \ud835\udcdc = mellin := by unfold mellin MellinTransform; simp_rw [smul_eq_mul, mul_comm]\n\nlemma MellinInverseTransform_eq (\u03c3 : \u211d) (f : \u2102 \u2192 \u2102) :\n    MellinInverseTransform f \u03c3 = mellinInv \u03c3 f := by\n  unfold mellinInv MellinInverseTransform VerticalIntegral' VerticalIntegral\n  beta_reduce; ext x\n  rw [\u2190 smul_assoc, smul_eq_mul (a' := I), div_mul]; simp\n\n/-%%\n\\begin{theorem}[MellinInversion]\\label{MellinInversion}\\lean{MellinInversion}\\leanok\nLet $f$ be a twice differentiable function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$, and\nlet $\\sigma$\nbe sufficiently large. Then\n$$f(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}\\mathcal{M}(f)(s)x^{-s}ds.$$\n\\end{theorem}\n\n%[Note: How ``nice''? Schwartz (on $(0,\\infty)$) is certainly enough. As we formalize\n%this, we can add whatever\n% conditions are necessary for the proof to go through.]\n%%-/\ntheorem MellinInversion (\u03c3 : \u211d) {f : \u211d \u2192 \u2102} {x : \u211d} (hx : 0 < x) (hf : MellinConvergent f \u03c3)\n    (hFf : VerticalIntegrable (mellin f) \u03c3) (hfx : ContinuousAt f x) :\n    MellinInverseTransform (\ud835\udcdc f) \u03c3 x = f x := by\n  rw [MellinTransform_eq, MellinInverseTransform_eq, mellin_inversion \u03c3 f hx hf hFf hfx]\n/-%%\n\\begin{proof}\\leanok\n\\uses{PartialIntegration, formulaLtOne, formulaGtOne, MellinTransform,\nMellinInverseTransform, PerronInverseMellin_gt, PerronInverseMellin_lt}\n%MellinInversion_aux1, MellinInversion_aux2, MellinInversion_aux3,\n%MellinInversion_aux4, }\nThe proof is from [Goldfeld-Kontorovich 2012].\nIntegrate by parts twice (assuming $f$ is twice differentiable, and all occurring\nintegrals converge absolutely, and\nboundary terms vanish).\n$$\n\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx = - \\int_0^\\infty f'(x)x^s\\frac{1}{s}dx\n= \\int_0^\\infty f''(x)x^{s+1}\\frac{1}{s(s+1)}dx.\n$$\nWe now have at least quadratic decay in $s$ of the Mellin transform. Inserting this\nformula into the inversion formula and Fubini-Tonelli (we now have absolute\nconvergence!) gives:\n$$\nRHS = \\frac{1}{2\\pi i}\\left(\\int_{(\\sigma)}\\int_0^\\infty\n  f''(t)t^{s+1}\\frac{1}{s(s+1)}dt\\right) x^{-s}ds\n$$\n$$\n= \\int_0^\\infty f''(t) t \\left( \\frac{1}{2\\pi i}\n\\int_{(\\sigma)}(t/x)^s\\frac{1}{s(s+1)}ds\\right) dt.\n$$\nApply the Perron formula to the inside:\n$$\n= \\int_x^\\infty f''(t) t \\left(1-\\frac{x}{t}\\right)dt\n= -\\int_x^\\infty f'(t) dt\n= f(x),\n$$\nwhere we integrated by parts (undoing the first partial integration), and finally\napplied the fundamental theorem of calculus (undoing the second).\n\\end{proof}\n%%-/\n\n\n/-%%\nFinally, we need Mellin Convolutions and properties thereof.\n\\begin{definition}[MellinConvolution]\\label{MellinConvolution}\\lean{MellinConvolution}\n\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. Then we define the\nMellin convolution of $f$ and $g$ to be the function $f\\ast g$ from $\\mathbb{R}_{>0}$\nto $\\mathbb{C}$ defined by\n$$(f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}.$$\n\\end{definition}\n%%-/\nnoncomputable def MellinConvolution (f g : \u211d \u2192 \ud835\udd42) (x : \u211d) : \ud835\udd42 :=\n  \u222b y in Ioi 0, f y * g (x / y) / y\n\n/-%%\nLet us start with a simple property of the Mellin convolution.\n\\begin{lemma}[MellinConvolutionSymmetric]\\label{MellinConvolutionSymmetric}\n\\lean{MellinConvolutionSymmetric}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{R}$ or $\\mathbb{C}$, for $x\\neq0$,\n$$\n  (f\\ast g)(x)=(g\\ast f)(x)\n  .\n$$\n\\end{lemma}\n%%-/\nlemma MellinConvolutionSymmetric (f g : \u211d \u2192 \ud835\udd42) {x : \u211d} (xpos: 0 < x) :\n    MellinConvolution f g x = MellinConvolution g f x := by\n  unfold MellinConvolution\n  calc\n    _ = \u222b y in Ioi 0, f (y * x) * g (1 / y) / y := ?_\n    _ = _ := ?_\n  \u00b7 rw [\u2190 integral_comp_mul_right_I0i_haar (fun y \u21a6 f y * g (x / y)) xpos]\n    simp [div_mul_cancel_right\u2080 <| ne_of_gt xpos]\n  \u00b7 convert (integral_comp_inv_I0i_haar fun y \u21a6 f (y * x) * g (1 / y)).symm using 3\n    rw [one_div_one_div, mul_comm, mul_comm_div, one_mul]\n/-%%\n\\begin{proof}\\leanok\n  \\uses{MellinConvolution}\n  By Definition \\ref{MellinConvolution},\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}\n  $$\n  in which we change variables to $z=x/y$:\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(x/z)g(z)\\frac{dz}{z}\n    =(g\\ast f)(x)\n    .\n  $$\n\\end{proof}\n%%-/\n\n/-%%\nThe Mellin transform of a convolution is the product of the Mellin transforms.\n\\begin{theorem}[MellinConvolutionTransform]\\label{MellinConvolutionTransform}\n\\lean{MellinConvolutionTransform}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ such that\n\\begin{equation}\n  (x,y)\\mapsto f(y)\\frac{g(x/y)}yx^{s-1}\n  \\label{eq:assm_integrable_Mconv}\n\\end{equation}\nis absolutely integrable on $[0,\\infty)^2$.\nThen\n$$\\mathcal{M}(f\\ast g)(s) = \\mathcal{M}(f)(s)\\mathcal{M}(g)(s).$$\n\\end{theorem}\n%%-/\nlemma MellinConvolutionTransform (f g : \u211d \u2192 \u2102) (s : \u2102)\n    (hf : IntegrableOn (fun x y \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)).uncurry\n      (Ioi 0 \u00d7\u02e2 Ioi 0)) :\n    \ud835\udcdc (MellinConvolution f g) s = \ud835\udcdc f s * \ud835\udcdc g s := by\n  dsimp [MellinTransform, MellinConvolution]\n  set f\u2081 : \u211d \u00d7 \u211d \u2192 \u2102 := fun \u27e8x, y\u27e9 \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, \u222b (y : \u211d) in Ioi 0, f\u2081 (x, y) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f\u2081 (x, y) := set_integral_integral_swap _ hf\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x / y) / \u2191y * \u2191x ^ (s - 1) := rfl\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x * y / y) / \u2191y * \u2191(x * y) ^ (s - 1) * y := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * (g x * \u2191x ^ (s - 1)) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * \u222b (x : \u211d) in Ioi 0, g x * \u2191x ^ (s - 1) := ?_\n    _ = _ := integral_mul_right _ _\n  <;> try (rw [set_integral_congr (by simp)]; intro y hy; simp only [ofReal_mul])\n  \u00b7 simp only [integral_mul_right]; rfl\n  \u00b7 simp only [integral_mul_right]\n    have := integral_comp_mul_right_Ioi (fun x \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)) 0 hy\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := slitPlane_ne_zero (Or.inl hy)\n    field_simp at this \u22a2\n    rw [this]\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro x hx\n    have y_ne_zero\u211d : y \u2260 0 := ne_of_gt (mem_Ioi.mp hy)\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := by exact_mod_cast y_ne_zero\u211d\n    field_simp [mul_cpow_ofReal_nonneg (LT.lt.le hx) (LT.lt.le hy)]\n    ring\n  \u00b7 apply integral_mul_left\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform,MellinConvolution}\nBy Definitions \\ref{MellinTransform} and \\ref{MellinConvolution}\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}\\frac{dy}ydx\n$$\nBy (\\ref{eq:assm_integrable_Mconv}) and Fubini's theorem,\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}dx\\frac{dy}y\n$$\nin which we change variables from $x$ to $z=x/y$:\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(z)y^{s-1}z^{s-1}dzdy\n$$\nwhich, by Definition \\ref{MellinTransform}, is\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\mathcal M(f)(s)\\mathcal M(g)(s)\n  .\n$$\n\n\\end{proof}\n%%-/\n\n/-%%\nLet $\\psi$ be a bumpfunction.\n\\begin{theorem}[SmoothExistence]\\label{SmoothExistence}\\lean{SmoothExistence}\\leanok\nThere exists a smooth (once differentiable would be enough), nonnegative ``bumpfunction'' $\\psi$,\n supported in $[1/2,2]$ with total mass one:\n$$\n\\int_0^\\infty \\psi(x)\\frac{dx}{x} = 1.\n$$\n\\end{theorem}\n%%-/\n\nattribute [- simp] one_div in\n\nlemma SmoothExistence : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n    \u03a8.support \u2286 Icc (1 / 2) 2 \u2227 \u222b x in Ici 0, \u03a8 x / x = 1 := by\n  suffices h : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n      \u03a8.support \u2286 Set.Icc (1 / 2) 2 \u2227 0 < \u222b x in Set.Ici 0, \u03a8 x / x by\n    rcases h with \u27e8\u03a8, h\u03a8, h\u03a8nonneg, h\u03a8supp, h\u03a8pos\u27e9\n    let c := (\u222b x in Ici 0, \u03a8 x / x)\n    use fun y \u21a6 \u03a8 y / c\n    refine \u27e8h\u03a8.div_const c, fun y \u21a6 div_nonneg (h\u03a8nonneg y) (le_of_lt h\u03a8pos), ?_, ?_\u27e9\n    \u00b7 rw [Function.support_div, Function.support_const (ne_of_lt h\u03a8pos).symm, inter_univ]\n      convert h\u03a8supp\n    \u00b7 simp only [div_right_comm _ c _, integral_div c, div_self <| ne_of_gt h\u03a8pos]\n\n  have := smooth_urysohn_support_Ioo (a := 1 / 2) (b := 1) (c := 3/2) (d := 2) (by linarith)\n    (by linarith)\n  rcases this with \u27e8\u03a8, h\u03a8ContDiff, _, h\u03a80, h\u03a81, h\u03a8Support\u27e9\n  use \u03a8, h\u03a8ContDiff\n  unfold indicator at h\u03a80 h\u03a81\n  simp only [mem_Icc, Pi.one_apply, Pi.le_def, mem_Ioo] at h\u03a80 h\u03a81\n  simp only [h\u03a8Support, subset_def, mem_Ioo, mem_Icc, and_imp]\n  split_ands\n  \u00b7 exact fun x \u21a6 le_trans (by simp [apply_ite]) (h\u03a80 x)\n  \u00b7 exact fun y hy hy' \u21a6 \u27e8by linarith, by linarith\u27e9\n  \u00b7 rw [integral_pos_iff_support_of_nonneg]\n    \u00b7 simp only [Function.support_div, measurableSet_Ici, Measure.restrict_apply', h\u03a8Support, Function.support_id]\n      have : (Ioo (1 / 2 : \u211d) 2 \u2229 (Iio 0 \u222a Ioi 0) \u2229 Ici 0) = Ioo (1 / 2) 2 := by\n        ext x\n        simp only [mem_inter_iff, mem_Ioo, mem_Ici, mem_Iio, mem_Ioi,\n          mem_union, not_lt, and_true, not_le]\n        constructor\n        \u00b7 exact fun h \u21a6 h.left.left\n        \u00b7 intro h\n          simp only [h, and_self, lt_or_lt_iff_ne, ne_eq, true_and]\n          constructor <;> linarith [h.left]\n      simp only [this, volume_Ioo, ENNReal.ofReal_pos, sub_pos, gt_iff_lt]\n      linarith\n    \u00b7 simp_rw [Pi.le_def, Pi.zero_apply]\n      intro y\n      by_cases h : y \u2208 Function.support \u03a8\n      . apply div_nonneg <| le_trans (by simp [apply_ite]) (h\u03a80 y)\n        rw [h\u03a8Support, mem_Ioo] at h\n        linarith [h.left]\n      . simp only [Function.mem_support, ne_eq, not_not] at h\n        simp [h]\n    \u00b7 have : (fun x \u21a6 \u03a8 x / x).support \u2286 Icc (1 / 2) 2 := by\n        rw [Function.support_div, h\u03a8Support]\n        apply subset_trans (by apply inter_subset_left) Ioo_subset_Icc_self\n      apply (integrableOn_iff_integrable_of_support_subset this).mp\n      apply ContinuousOn.integrableOn_compact isCompact_Icc\n      apply ContinuousOn.div h\u03a8ContDiff.continuous.continuousOn continuousOn_id ?_\n      simp only [mem_Icc, ne_eq, and_imp, id_eq]\n      intros\n      linarith\n/-%%\n\\begin{proof}\\leanok\n\\uses{smooth-ury}\nSame idea as Urysohn-type argument.\n\\end{proof}\n%%-/\n\nlemma mem_within_strip (\u03c3\u2081 \u03c3\u2082 : \u211d) :\n  {s : \u2102 | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} \u2208 \ud835\udcdf {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} := by simp\n\nlemma MellinOfPsi_aux {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {s : \u2102} (hs : s \u2260 0) :\n    \u222b (x : \u211d) in Ioi 0, (\u03a8 x) * (x : \u2102) ^ (s - 1) =\n    - (1 / s) * \u222b (x : \u211d) in Ioi 0, (deriv \u03a8 x) * (x : \u2102) ^ s := by\n  let g {s : \u2102} := fun (x : \u211d)  \u21a6 x ^ s / s\n  have gderiv {s : \u2102} (hs : s \u2260 0) {x: \u211d} (hx : x \u2208 Ioi 0) :\n      deriv g x = x ^ (s - 1) := by\n    have := HasDerivAt.cpow_const (c := s) (hasDerivAt_id (x : \u2102)) (Or.inl hx)\n    simp_rw [mul_one, id_eq] at this\n    rw [deriv_div_const, deriv.comp_ofReal (e := fun x \u21a6 x ^ s)]\n    \u00b7 rw [this.deriv, mul_div_right_comm, div_self hs, one_mul]\n    \u00b7 apply hasDerivAt_deriv_iff.mp\n      simp only [this.deriv, this]\n  calc\n    _ =  \u222b (x : \u211d) in Ioi 0, \u2191(\u03a8 x) * deriv (@g s) x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv (fun x \u21a6 \u2191(\u03a8 x)) x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * x ^ s / s := by simp only [mul_div, g]\n    _ = _ := ?_\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro _ hx\n    simp only [gderiv hs hx]\n  \u00b7 apply PartialIntegration_of_support_in_Icc (\u03a8 \u00b7) g\n      (a := 1 / 2) (b := 2) (by norm_num) (by norm_num)\n    \u00b7 simpa only [Function.support_subset_iff, ne_eq, ofReal_eq_zero]\n    \u00b7 exact (Differentiable.ofReal_comp_iff.mpr (diff\u03a8.differentiable (by norm_num))).differentiableOn\n    \u00b7 refine DifferentiableOn.div_const ?_ s\n      intro a ha\n      refine DifferentiableAt.differentiableWithinAt ?_\n      apply DifferentiableAt.comp_ofReal (e := fun x \u21a6 x ^ s)\n      apply DifferentiableAt.cpow differentiableAt_id' <| differentiableAt_const s\n      exact Or.inl ha\n    \u00b7 simp only [deriv.ofReal_comp']\n      apply Continuous.continuousOn\n      apply Continuous.comp (g := ofReal') continuous_ofReal <| diff\u03a8.continuous_deriv (by norm_num)\n    \u00b7 apply ContinuousOn.congr (f := fun (x : \u211d) \u21a6 (x : \u2102) ^ (s - 1)) ?_ fun x hx \u21a6 gderiv hs hx\n      refine ContinuousOn.cpow ?_ continuousOn_const (by simp)\n      exact Continuous.continuousOn (by continuity)\n  \u00b7 congr; funext; congr\n    apply (hasDerivAt_deriv_iff.mpr ?_).ofReal_comp.deriv\n    exact diff\u03a8.contDiffAt.differentiableAt (by norm_num)\n  \u00b7 simp only [neg_mul, neg_inj]\n    conv => lhs; rhs; intro; rw [\u2190 mul_one_div, mul_comm]\n    rw [integral_mul_left]\n\n/-%%\nThe $\\psi$ function has Mellin transform $\\mathcal{M}(\\psi)(s)$ which is entire and decays (at\nleast) like $1/|s|$.\n\\begin{theorem}[MellinOfPsi]\\label{MellinOfPsi}\\lean{MellinOfPsi}\\leanok\nThe Mellin transform of $\\psi$ is\n$$\\mathcal{M}(\\psi)(s) =  O\\left(\\frac{1}{|s|}\\right),$$\nas $|s|\\to\\infty$ with $\\sigma_1 \\le \\Re(s) \\le \\sigma_2$.\n\\end{theorem}\n\n[Of course it decays faster than any power of $|s|$, but it turns out that we will just need one\npower.]\n%%-/\nlemma MellinOfPsi {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {\u03c3\u2081 \u03c3\u2082 : \u211d} (\u03c3\u2081pos : 0 < \u03c3\u2081) :\n    (fun s \u21a6 \u2016\ud835\udcdc (\u03a8 \u00b7) s\u2016)\n    =O[Filter.principal {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082}]\n      fun s \u21a6 1 / \u2016s\u2016 := by\n  let f := fun (x : \u211d) \u21a6 \u2016deriv \u03a8 x\u2016\n  have cont : ContinuousOn f (Icc (1 / 2) 2) :=\n    (Continuous.comp (by continuity) <| diff\u03a8.continuous_deriv (by norm_num)).continuousOn\n  obtain \u27e8a, _, max\u27e9 := isCompact_Icc.exists_isMaxOn (f := f) (by norm_num) cont\n  rw [Asymptotics.isBigO_iff]\n  use f a * 2 ^ \u03c3\u2082 * (3 / 2)\n  filter_upwards [mem_within_strip \u03c3\u2081 \u03c3\u2082] with s hs\n  have s_ne_zero: s \u2260 0 := fun h \u21a6 by linarith [zero_re \u25b8 h \u25b8 hs.1]\n  simp only [MellinTransform, f, MellinOfPsi_aux diff\u03a8 supp\u03a8 s_ne_zero, norm_norm, norm_mul]\n  conv => rhs; rw [mul_comm]\n  gcongr; simp\n  calc\n    _ \u2264 \u222b (x : \u211d) in Ioi 0, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016 := ?_\n    _ = \u222b (x : \u211d) in Icc (1 / 2) 2, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016 := ?_\n    _ \u2264 \u2016\u222b (x : \u211d) in Icc (1 / 2) 2, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016\u2016 := le_abs_self _\n    _ \u2264 _ := ?_\n  \u00b7 simp_rw [norm_integral_le_integral_norm]\n  \u00b7 apply SetIntegral.integral_eq_integral_inter_of_support_subset_Icc\n    \u00b7 simp only [Function.support_abs, Function.support_mul, Function.support_ofReal]\n      apply subset_trans (by apply inter_subset_left) <| Function.support_deriv_subset_Icc supp\u03a8\n    \u00b7 exact (Icc_subset_Ioi_iff (by norm_num)).mpr (by norm_num)\n  \u00b7 have := intervalIntegral.norm_integral_le_of_norm_le_const' (C := f a * 2 ^ \u03c3\u2082)\n      (f := fun x \u21a6 f x * \u2016(x : \u2102) ^ s\u2016) (a := (1 / 2 : \u211d)) ( b := 2) (by norm_num) ?_\n    \u00b7 simp only [Real.norm_eq_abs, Complex.norm_eq_abs, abs_ofReal, map_mul] at this \u22a2\n      rwa [(by norm_num: |(2 : \u211d) - 1 / 2| = 3 / 2),\n          intervalIntegral.integral_of_le (by norm_num), \u2190 integral_Icc_eq_integral_Ioc] at this\n    \u00b7 intro x hx;\n      have f_bound := isMaxOn_iff.mp max x hx\n      have pow_bound : \u2016(x : \u2102) ^ s\u2016 \u2264 2 ^ \u03c3\u2082 := by\n        rw [Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos (by linarith [mem_Icc.mp hx])]\n        have xpos : 0 \u2264 x := by linarith [(mem_Icc.mp hx).1]\n        have h := rpow_le_rpow xpos (mem_Icc.mp hx).2 (by linarith : 0 \u2264 s.re)\n        exact le_trans h <| rpow_le_rpow_of_exponent_le (by norm_num) hs.2\n      convert mul_le_mul f_bound pow_bound (norm_nonneg _) ?_ using 1 <;> simp [f]\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform, SmoothExistence}\nIntegrate by parts:\n$$\n\\left|\\int_0^\\infty \\psi(x)x^s\\frac{dx}{x}\\right| =\n\\left|-\\int_0^\\infty \\psi'(x)\\frac{x^{s}}{s}dx\\right|\n$$\n$$\n\\le \\frac{1}{|s|} \\int_{1/2}^2|\\psi'(x)|x^{\\Re(s)}dx.\n$$\nSince $\\Re(s)$ is bounded, the right-hand side is bounded by a\nconstant times $1/|s|$.\n\\end{proof}\n%%-/\n\n/-%%\nWe can make a delta spike out of this bumpfunction, as follows.\n\\begin{definition}[DeltaSpike]\\label{DeltaSpike}\\lean{DeltaSpike}\\leanok\n\\uses{SmoothExistence}\nLet $\\psi$ be a bumpfunction supported in $[1/2,2]$. Then for any $\\epsilon>0$, we define the\ndelta spike $\\psi_\\epsilon$ to be the function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\psi_\\epsilon(x) = \\frac{1}{\\epsilon}\\psi\\left(x^{\\frac{1}{\\epsilon}}\\right).$$\n\\end{definition}\n%%-/\n\nnoncomputable def DeltaSpike (\u03a8 : \u211d \u2192 \u211d) (\u03b5 : \u211d) : \u211d \u2192 \u211d :=\n  fun x \u21a6 \u03a8 (x ^ (1 / \u03b5)) / \u03b5\n\n/-%%\nThis spike still has mass one:\n\\begin{lemma}[DeltaSpikeMass]\\label{DeltaSpikeMass}\\lean{DeltaSpikeMass}\\leanok\nFor any $\\epsilon>0$, we have\n$$\\int_0^\\infty \\psi_\\epsilon(x)\\frac{dx}{x} = 1.$$\n\\end{lemma}\n%%-/\n\nlemma DeltaSpikeMass {\u03a8 : \u211d \u2192 \u211d} (mass_one: \u222b x in Ioi 0, \u03a8 x / x = 1) {\u03b5 : \u211d}\n    (\u03b5pos : 0 < \u03b5) : \u222b x in Ioi 0, ((DeltaSpike \u03a8 \u03b5) x) / x = 1 :=\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, (|1/\u03b5| * x ^ (1 / \u03b5 - 1)) \u2022\n      ((fun z \u21a6 (\u03a8 z) / z) (x ^ (1 / \u03b5))) := by\n      apply set_integral_congr_ae measurableSet_Ioi\n      filter_upwards with x hx\n      simp only [mem_Ioi, smul_eq_mul, abs_of_pos (one_div_pos.mpr \u03b5pos)]\n      symm; calc\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / x ^ (1 / \u03b5)) * x ^ (1 / \u03b5 - 1) * (1 / \u03b5) := by ring\n        _ = _ := by rw [rpow_sub hx, rpow_one]\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / x ^ (1 / \u03b5) * x ^ (1 / \u03b5) / x) * (1/ \u03b5) := by ring\n        _ = _ := by rw [div_mul_cancel\u2080 _ (ne_of_gt (rpow_pos_of_pos hx (1/\u03b5)))]\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / \u03b5 / x) := by ring\n    _ = 1 := by\n      rw [integral_comp_rpow_Ioi (fun z \u21a6 (\u03a8 z) / z), \u2190 mass_one]\n      simp only [ne_eq, div_eq_zero_iff, one_ne_zero, \u03b5pos.ne', or_self, not_false_eq_true]\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{DeltaSpike}\nSubstitute $y=x^{1/\\epsilon}$, and use the fact that $\\psi$ has mass one, and that $dx/x$ is Haar\nmeasure.\n\\end{proof}\n%%-/\n\nlemma DeltaSpikeSupport_aux {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2):\n    (fun x \u21a6 if x < 0 then 0 else DeltaSpike \u03a8 \u03b5 x).support \u2286 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) := by\n  unfold DeltaSpike\n  simp only [one_div, Function.support_subset_iff, ne_eq, ite_eq_left_iff, not_lt, div_eq_zero_iff,\n    not_forall, exists_prop, mem_Icc, and_imp]\n  intro x hx h; push_neg at h\n  have := supp\u03a8 <| Function.mem_support.mpr h.1\n  simp only [one_div, mem_Icc] at this\n  have hl := (le_rpow_inv_iff_of_pos (by norm_num) hx \u03b5pos).mp this.1\n  rw [inv_rpow (by norm_num) \u03b5, \u2190 rpow_neg (by norm_num)] at hl\n  refine \u27e8hl, (rpow_inv_le_iff_of_pos ?_ (by norm_num) \u03b5pos).mp this.2\u27e9\n  linarith [(by apply rpow_nonneg (by norm_num) : 0 \u2264 (2 : \u211d) ^ (-\u03b5))]\n\n", "theoremStatement": "lemma DeltaSpikeSupport' {\u03a8 : \u211d \u2192 \u211d} {\u03b5 x : \u211d} (\u03b5pos : 0 < \u03b5) (xnonneg : 0 \u2264 x)\n    (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2) :\n    DeltaSpike \u03a8 \u03b5 x \u2260 0 \u2192 x \u2208 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) ", "theoremName": "DeltaSpikeSupport'", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "5acfe03684a7f39bbed879fea9a994fc5c5d55cf", "date": "2024-03-31"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/MellinCalculus.lean", "module": "PrimeNumberTheoremAnd.MellinCalculus", "jsonFile": "PrimeNumberTheoremAnd.MellinCalculus.jsonl", "positionMetadata": {"lineInFile": 937, "tokenPositionInFile": 40184, "theoremPositionInFile": 61}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 62, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", 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"Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  intro h\n  have : (fun x \u21a6 if x < 0 then 0 else DeltaSpike \u03a8 \u03b5 x) x = DeltaSpike \u03a8 \u03b5 x := by simp [xnonneg]\n  rw [\u2190 this] at h\n  exact (Function.support_subset_iff.mp <| DeltaSpikeSupport_aux \u03b5pos supp\u03a8) _ h", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 214}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]\n\nlemma Filter.BigO_zero_atTop_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[atTop] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioi_mem_atTop b] with c hc; replace hc := mem_Ioi.mp hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h]\n\n-- steal coerction lemmas from EulerProducts.Auxiliary because of build issues, and add new ones\nnamespace Complex\n-- see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Differentiability.20of.20the.20natural.20map.20.E2.84.9D.20.E2.86.92.20.E2.84.82/near/418095234\n\nlemma hasDerivAt_ofReal (x : \u211d) : HasDerivAt ofReal' 1 x :=\n  HasDerivAt.ofReal_comp <| hasDerivAt_id x\n\nlemma deriv_ofReal (x : \u211d) : deriv ofReal' x = 1 :=\n  (hasDerivAt_ofReal x).deriv\n\nlemma differentiableAt_ofReal (x : \u211d) : DifferentiableAt \u211d ofReal' x :=\n  (hasDerivAt_ofReal x).differentiableAt\n\nlemma differentiable_ofReal : Differentiable \u211d ofReal' :=\n  ofRealCLM.differentiable\n\nend Complex\n\nlemma DifferentiableAt.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    DifferentiableAt \u211d (fun x : \u211d \u21a6 e x) z :=\n  hf.hasDerivAt.comp_ofReal.differentiableAt\n\nlemma Differentiable.comp_ofReal {e : \u2102 \u2192 \u2102} (h : Differentiable \u2102 e) :\n    Differentiable \u211d (fun x : \u211d \u21a6 e x) :=\n  fun _ \u21a6 h.differentiableAt.comp_ofReal\n\nlemma DifferentiableAt.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} (hf : DifferentiableAt \u211d f z) :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z :=\n  hf.hasDerivAt.ofReal_comp.differentiableAt\n\nlemma Differentiable.ofReal_comp {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) :=\n  fun _ \u21a6 hf.differentiableAt.ofReal_comp\n\nopen Complex ContinuousLinearMap in\nlemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} {u : \u2102}\n    (hf : HasDerivAt (fun y \u21a6 (f y : \u2102)) u z) :\n    \u2203 u' : \u211d, u = u' \u2227 HasDerivAt f u' z := by\n  lift u to \u211d\n  \u00b7 have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv\n    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H\n    rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H\n  refine \u27e8u, rfl, ?_\u27e9\n  convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt\n  rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]\n\nlemma DifferentiableAt.ofReal_comp_iff {z : \u211d} {f : \u211d \u2192 \u211d} :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z \u2194 DifferentiableAt \u211d f z := by\n  refine \u27e8fun H \u21a6 ?_, ofReal_comp\u27e9\n  obtain \u27e8u, _, hu\u2082\u27e9 := H.hasDerivAt.of_hasDerivAt_ofReal_comp\n  exact HasDerivAt.differentiableAt hu\u2082\n\nlemma Differentiable.ofReal_comp_iff {f : \u211d \u2192 \u211d} :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) \u2194 Differentiable \u211d f :=\n  forall_congr' fun _ \u21a6 DifferentiableAt.ofReal_comp_iff\n\nlemma deriv.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} :\n    deriv (fun (y : \u211d) \u21a6 (f y : \u2102)) z = deriv f z := by\n  by_cases hf : DifferentiableAt \u211d f z\n  \u00b7 exact hf.hasDerivAt.ofReal_comp.deriv\n  \u00b7 have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf\n    rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt <| hf',\n      Complex.ofReal_zero]\n\nlemma deriv.ofReal_comp' {f : \u211d \u2192 \u211d} :\n    deriv (fun x : \u211d \u21a6 (f x : \u2102)) = (fun x \u21a6 ((deriv f) x : \u2102)) :=\n  funext fun _ \u21a6 deriv.ofReal_comp\n\nlemma deriv.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    deriv (fun x : \u211d \u21a6 e x) z = deriv e z :=\n  hf.hasDerivAt.comp_ofReal.deriv\n\nlemma deriv.comp_ofReal' {e : \u2102 \u2192 \u2102} (hf : Differentiable \u2102 e) :\n    deriv (fun x : \u211d \u21a6 e x) = fun (x : \u211d) \u21a6 deriv e x :=\n  funext fun _ \u21a6 deriv.comp_ofReal (hf.differentiableAt)\n\n/-%%\n\\begin{lemma}[PartialIntegration]\\label{PartialIntegration}\\lean{PartialIntegration}\\leanok\nLet $f, g$ be once differentiable functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ so that $fg'$\nand $f'g$ are both integrable, and $f*g (x)\\to 0$ as $x\\to 0^+,\\infty$.\nThen\n$$\n\\int_0^\\infty f(x)g'(x) dx = -\\int_0^\\infty f'(x)g(x)dx.\n$$\n\\end{lemma}\n%%-/\n/-- *Need differentiability, and decay at `0` and `\u221e`* -/\nlemma PartialIntegration (f g : \u211d \u2192 \u2102)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0))\n    (gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0))\n    (lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  simpa using integral_Ioi_mul_deriv_eq_deriv_mul\n    (fun x hx \u21a6 fDiff.hasDerivAt (Ioi_mem_nhds hx))\n    (fun x hx \u21a6 gDiff.hasDerivAt (Ioi_mem_nhds hx))\n    fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\nlemma PartialIntegration_of_support_in_Icc {a b : \u211d} (f g : \u211d \u2192 \u2102) (ha : 0 < a) (h : a \u2264 b)\n    (fSupp : f.support \u2286 Set.Icc a b)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fderivCont : ContinuousOn (deriv f) (Ioi 0))\n    (gderivCont : ContinuousOn (deriv g) (Ioi 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  have Icc_sub : Icc a b \u2286 Ioi 0 := (Icc_subset_Ioi_iff h).mpr ha\n  have fderivSupp := Function.support_deriv_subset_Icc fSupp\n  have fgSupp : (f * g).support \u2286 Icc a b := Function.support_mul_subset_of_subset fSupp\n  have fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fDiff.continuousOn.mono Icc_sub\n    \u00b7 exact gderivCont.mono Icc_sub\n  have gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fderivSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fderivCont.mono Icc_sub\n    \u00b7 exact gDiff.continuousOn.mono Icc_sub\n  have lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0) := TendstoAtZero_of_support_in_Icc (f * g) ha fgSupp\n  have lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0) := TendstoAtTop_of_support_in_Icc (f * g) fgSupp\n  apply PartialIntegration f g fDiff gDiff fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n\n/-%%\nIn this section, we define the Mellin transform (already in Mathlib, thanks to David Loeffler),\nprove its inversion formula, and\nderive a number of important properties of some special functions and bumpfunctions.\n\nDef: (Already in Mathlib)\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be the function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\n[Note: My preferred way to think about this is that we are integrating over the multiplicative\ngroup $\\mathbb{R}_{>0}$, multiplying by a (not necessarily unitary!) character $|\\cdot|^s$, and\nintegrating with respect to the invariant Haar measure $dx/x$. This is very useful in the kinds\nof calculations carried out below. But may be more difficult to formalize as things now stand. So\nwe might have clunkier calculations, which ``magically'' turn out just right - of course they're\nexplained by the aforementioned structure...]\n\n%%-/\n\n\n/-%%\n\\begin{definition}[MellinTransform]\\label{MellinTransform}\\lean{MellinTransform}\\leanok\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be\nthe function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\\end{definition}\n[Note: already exists in Mathlib, with some good API.]\n%%-/\nnoncomputable def MellinTransform (f : \u211d \u2192 \u2102) (s : \u2102) : \u2102 :=\n  \u222b x in Ioi 0, f x * x ^ (s - 1)\n\nlocal notation (name := mellintransform) \"\ud835\udcdc\" => MellinTransform\n/-%%\n\\begin{definition}[MellinInverseTransform]\\label{MellinInverseTransform}\n\\lean{MellinInverseTransform}\\leanok\nLet $F$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$. We define the Mellin inverse transform of\n$F$ to be the function $\\mathcal{M}^{-1}(F)$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}^{-1}(F)(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}F(s)x^{-s}ds,$$\nwhere $\\sigma$ is sufficiently large (say $\\sigma>2$).\n\\end{definition}\n%%-/\nnoncomputable def MellinInverseTransform (F : \u2102 \u2192 \u2102) (\u03c3 : \u211d) (x : \u211d) : \u2102 :=\n  VerticalIntegral' (fun s \u21a6 x ^ (-s) * F s) \u03c3\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_lt]\\label{PerronInverseMellin_lt}\\lean{PerronInverseMellin_lt}\n\\leanok\nLet $0 < t < x$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function\n$$F: s\\mapsto t^s/s(s+1)$$ is equal to\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds\n= 0.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_lt {t x : \u211d} (tpos : 0 < t) (t_lt_x : t < x) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 0 := by\n  dsimp [MellinInverseTransform, VerticalIntegral']\n  have xpos : 0 < x := lt_trans tpos t_lt_x\n  simp only [one_div, mul_inv_rev, inv_I, neg_mul, neg_eq_zero, mul_eq_zero, I_ne_zero,\n    inv_eq_zero, ofReal_eq_zero, pi_ne_zero, OfNat.ofNat_ne_zero, or_self, false_or]\n  convert Perron.formulaLtOne (div_pos tpos xpos) ((div_lt_one xpos).mpr t_lt_x) \u03c3pos using 2\n  ext1 s\n  convert Perron.f_mul_eq_f tpos xpos s using 1\n  ring\n/-%%\n\\begin{proof}\\leanok\n\\uses{Perron.formulaLtOne}\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_gt]\\label{PerronInverseMellin_gt}\\lean{PerronInverseMellin_gt}\n\\leanok\nLet $0 < x < t$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function is equal\nto\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds = 1 - x / t.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_gt {t x : \u211d} (xpos : 0 < x) (x_lt_t : x < t) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 1 - x / t := by\n  dsimp [MellinInverseTransform]\n  have tpos : 0 < t := by linarith\n  have txinv_gtOne : 1 < t / x := (one_lt_div xpos).mpr x_lt_t\n  rw [\u2190 smul_eq_mul]\n  convert Perron.formulaGtOne txinv_gtOne \u03c3pos using 2\n  \u00b7 congr\n    ext1 s\n    convert Perron.f_mul_eq_f tpos xpos s using 1\n    ring\n  \u00b7 field_simp\n/-%%\n\\begin{proof}\n\\uses{Perron.formulaGtOne}\\leanok\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-% ** Wrong delimiters on purpose **\nUnnecessary lemma:\n%\\begin{lemma}[MellinInversion_aux1]\\label{MellinInversion_aux1}\\lean{MellinInversion_aux1}\\leanok\nLet $f$ be differentiable on $(0,\\infty)$, and assume that $f(x)x^s\\to 0$ as $x\\to 0$, and that\n$f(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f(x)x^s\\frac{dx}{x} = \\frac{1}{s}\\int_0^\\infty f'(x)x^{s} dx.\n$$\n%\\end{lemma}\n%-/\nlemma MellinInversion_aux1 {f : \u211d \u2192 \u2102} {s : \u2102} (s_ne_zero : s \u2260 0)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * x ^ s / x = - \u222b x in Ioi 0, (deriv f x) * x ^ s / s := by\n  sorry\n\n/-% ** Wrong delimiters on purpose **\n\\begin{proof}\n\\uses{PartialIntegration}\nPartial integration.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux2]\\label{MellinInversion_aux2}\\lean{MellinInversion_aux2}\\leanok\nLet $f$ be twice differentiable on $(0,\\infty)$, and assume that $f'(x)x^s\\to 0$ as $x\\to 0$, and\nthat $f'(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f'(x)x^{s} dx = -\\int_0^\\infty f''(x)x^{s+1}\\frac{1}{(s+1)}dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux2 {f : \u211d \u2192 \u2102} (s : \u2102) (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (fDiff2 : DifferentiableOn \u211d (deriv f) (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 deriv f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 deriv f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, (deriv f x) * x ^ s =\n      -\u222b x in Ioi 0, (deriv (deriv f) x) * x ^ (s + 1) / (s + 1) := by\n  sorry\n/-%\n\\begin{proof}\n\\uses{PartialIntegration, MellinInversion_aux1}\nPartial integration. (Apply Lemma \\ref{MellinInversion_aux1} to the function $f'$ and $s+1$.)\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux3]%\\label{MellinInversion_aux3}\\lean{MellinInversion_aux3}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen the map  $(x,s) \\mapsto f(x)x^s/(s(s+1))$ is absolutely integrable on\n$(0,\\infty)\\times\\{\\Re s = \\sigma\\}$ for any $\\sigma>0$.\n\\end{lemma}\n%-/\nlemma MellinInversion_aux3 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    IntegrableOn (fun (\u27e8x, t\u27e9 : \u211d \u00d7 \u211d) \u21a6 f x * x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * ((\u03c3 + t * I) + 1)))\n      ((Ioi 0).prod (univ : Set \u211d)) := by\n  sorry\n/-%\n\\begin{proof}\nPut absolute values and estimate.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux4]%\\label{MellinInversion_aux4}\\lean{MellinInversion_aux4}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen we can interchange orders of integration\n$$\n\\int_{(\\sigma)}\\int_0^\\infty f(x)x^{s+1}\\frac{1}{s(s+1)}dx ds =\n\\int_0^\\infty\n\\int_{(\\sigma)}f(x)x^{s+1}\\frac{1}{s(s+1)}ds dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux4 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    VerticalIntegral (fun s \u21a6 \u222b x in Ioi 0, f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 =\n      \u222b x in Ioi 0, VerticalIntegral (fun s \u21a6 f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 := by\n  sorry -- `MeasureTheory.integral_prod` and `MeasureTheory.integral_swap` should be useful here\n/-%\n\\begin{proof}\n\\uses{MellinInversion_aux3}\nFubini-Tonelli.\n\\end{proof}\n%-/\n\nlemma MellinTransform_eq : \ud835\udcdc = mellin := by unfold mellin MellinTransform; simp_rw [smul_eq_mul, mul_comm]\n\nlemma MellinInverseTransform_eq (\u03c3 : \u211d) (f : \u2102 \u2192 \u2102) :\n    MellinInverseTransform f \u03c3 = mellinInv \u03c3 f := by\n  unfold mellinInv MellinInverseTransform VerticalIntegral' VerticalIntegral\n  beta_reduce; ext x\n  rw [\u2190 smul_assoc, smul_eq_mul (a' := I), div_mul]; simp\n\n/-%%\n\\begin{theorem}[MellinInversion]\\label{MellinInversion}\\lean{MellinInversion}\\leanok\nLet $f$ be a twice differentiable function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$, and\nlet $\\sigma$\nbe sufficiently large. Then\n$$f(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}\\mathcal{M}(f)(s)x^{-s}ds.$$\n\\end{theorem}\n\n%[Note: How ``nice''? Schwartz (on $(0,\\infty)$) is certainly enough. As we formalize\n%this, we can add whatever\n% conditions are necessary for the proof to go through.]\n%%-/\ntheorem MellinInversion (\u03c3 : \u211d) {f : \u211d \u2192 \u2102} {x : \u211d} (hx : 0 < x) (hf : MellinConvergent f \u03c3)\n    (hFf : VerticalIntegrable (mellin f) \u03c3) (hfx : ContinuousAt f x) :\n    MellinInverseTransform (\ud835\udcdc f) \u03c3 x = f x := by\n  rw [MellinTransform_eq, MellinInverseTransform_eq, mellin_inversion \u03c3 f hx hf hFf hfx]\n/-%%\n\\begin{proof}\\leanok\n\\uses{PartialIntegration, formulaLtOne, formulaGtOne, MellinTransform,\nMellinInverseTransform, PerronInverseMellin_gt, PerronInverseMellin_lt}\n%MellinInversion_aux1, MellinInversion_aux2, MellinInversion_aux3,\n%MellinInversion_aux4, }\nThe proof is from [Goldfeld-Kontorovich 2012].\nIntegrate by parts twice (assuming $f$ is twice differentiable, and all occurring\nintegrals converge absolutely, and\nboundary terms vanish).\n$$\n\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx = - \\int_0^\\infty f'(x)x^s\\frac{1}{s}dx\n= \\int_0^\\infty f''(x)x^{s+1}\\frac{1}{s(s+1)}dx.\n$$\nWe now have at least quadratic decay in $s$ of the Mellin transform. Inserting this\nformula into the inversion formula and Fubini-Tonelli (we now have absolute\nconvergence!) gives:\n$$\nRHS = \\frac{1}{2\\pi i}\\left(\\int_{(\\sigma)}\\int_0^\\infty\n  f''(t)t^{s+1}\\frac{1}{s(s+1)}dt\\right) x^{-s}ds\n$$\n$$\n= \\int_0^\\infty f''(t) t \\left( \\frac{1}{2\\pi i}\n\\int_{(\\sigma)}(t/x)^s\\frac{1}{s(s+1)}ds\\right) dt.\n$$\nApply the Perron formula to the inside:\n$$\n= \\int_x^\\infty f''(t) t \\left(1-\\frac{x}{t}\\right)dt\n= -\\int_x^\\infty f'(t) dt\n= f(x),\n$$\nwhere we integrated by parts (undoing the first partial integration), and finally\napplied the fundamental theorem of calculus (undoing the second).\n\\end{proof}\n%%-/\n\n\n/-%%\nFinally, we need Mellin Convolutions and properties thereof.\n\\begin{definition}[MellinConvolution]\\label{MellinConvolution}\\lean{MellinConvolution}\n\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. Then we define the\nMellin convolution of $f$ and $g$ to be the function $f\\ast g$ from $\\mathbb{R}_{>0}$\nto $\\mathbb{C}$ defined by\n$$(f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}.$$\n\\end{definition}\n%%-/\nnoncomputable def MellinConvolution (f g : \u211d \u2192 \ud835\udd42) (x : \u211d) : \ud835\udd42 :=\n  \u222b y in Ioi 0, f y * g (x / y) / y\n\n/-%%\nLet us start with a simple property of the Mellin convolution.\n\\begin{lemma}[MellinConvolutionSymmetric]\\label{MellinConvolutionSymmetric}\n\\lean{MellinConvolutionSymmetric}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{R}$ or $\\mathbb{C}$, for $x\\neq0$,\n$$\n  (f\\ast g)(x)=(g\\ast f)(x)\n  .\n$$\n\\end{lemma}\n%%-/\nlemma MellinConvolutionSymmetric (f g : \u211d \u2192 \ud835\udd42) {x : \u211d} (xpos: 0 < x) :\n    MellinConvolution f g x = MellinConvolution g f x := by\n  unfold MellinConvolution\n  calc\n    _ = \u222b y in Ioi 0, f (y * x) * g (1 / y) / y := ?_\n    _ = _ := ?_\n  \u00b7 rw [\u2190 integral_comp_mul_right_I0i_haar (fun y \u21a6 f y * g (x / y)) xpos]\n    simp [div_mul_cancel_right\u2080 <| ne_of_gt xpos]\n  \u00b7 convert (integral_comp_inv_I0i_haar fun y \u21a6 f (y * x) * g (1 / y)).symm using 3\n    rw [one_div_one_div, mul_comm, mul_comm_div, one_mul]\n/-%%\n\\begin{proof}\\leanok\n  \\uses{MellinConvolution}\n  By Definition \\ref{MellinConvolution},\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}\n  $$\n  in which we change variables to $z=x/y$:\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(x/z)g(z)\\frac{dz}{z}\n    =(g\\ast f)(x)\n    .\n  $$\n\\end{proof}\n%%-/\n\n/-%%\nThe Mellin transform of a convolution is the product of the Mellin transforms.\n\\begin{theorem}[MellinConvolutionTransform]\\label{MellinConvolutionTransform}\n\\lean{MellinConvolutionTransform}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ such that\n\\begin{equation}\n  (x,y)\\mapsto f(y)\\frac{g(x/y)}yx^{s-1}\n  \\label{eq:assm_integrable_Mconv}\n\\end{equation}\nis absolutely integrable on $[0,\\infty)^2$.\nThen\n$$\\mathcal{M}(f\\ast g)(s) = \\mathcal{M}(f)(s)\\mathcal{M}(g)(s).$$\n\\end{theorem}\n%%-/\nlemma MellinConvolutionTransform (f g : \u211d \u2192 \u2102) (s : \u2102)\n    (hf : IntegrableOn (fun x y \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)).uncurry\n      (Ioi 0 \u00d7\u02e2 Ioi 0)) :\n    \ud835\udcdc (MellinConvolution f g) s = \ud835\udcdc f s * \ud835\udcdc g s := by\n  dsimp [MellinTransform, MellinConvolution]\n  set f\u2081 : \u211d \u00d7 \u211d \u2192 \u2102 := fun \u27e8x, y\u27e9 \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, \u222b (y : \u211d) in Ioi 0, f\u2081 (x, y) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f\u2081 (x, y) := set_integral_integral_swap _ hf\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x / y) / \u2191y * \u2191x ^ (s - 1) := rfl\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x * y / y) / \u2191y * \u2191(x * y) ^ (s - 1) * y := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * (g x * \u2191x ^ (s - 1)) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * \u222b (x : \u211d) in Ioi 0, g x * \u2191x ^ (s - 1) := ?_\n    _ = _ := integral_mul_right _ _\n  <;> try (rw [set_integral_congr (by simp)]; intro y hy; simp only [ofReal_mul])\n  \u00b7 simp only [integral_mul_right]; rfl\n  \u00b7 simp only [integral_mul_right]\n    have := integral_comp_mul_right_Ioi (fun x \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)) 0 hy\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := slitPlane_ne_zero (Or.inl hy)\n    field_simp at this \u22a2\n    rw [this]\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro x hx\n    have y_ne_zero\u211d : y \u2260 0 := ne_of_gt (mem_Ioi.mp hy)\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := by exact_mod_cast y_ne_zero\u211d\n    field_simp [mul_cpow_ofReal_nonneg (LT.lt.le hx) (LT.lt.le hy)]\n    ring\n  \u00b7 apply integral_mul_left\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform,MellinConvolution}\nBy Definitions \\ref{MellinTransform} and \\ref{MellinConvolution}\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}\\frac{dy}ydx\n$$\nBy (\\ref{eq:assm_integrable_Mconv}) and Fubini's theorem,\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}dx\\frac{dy}y\n$$\nin which we change variables from $x$ to $z=x/y$:\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(z)y^{s-1}z^{s-1}dzdy\n$$\nwhich, by Definition \\ref{MellinTransform}, is\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\mathcal M(f)(s)\\mathcal M(g)(s)\n  .\n$$\n\n\\end{proof}\n%%-/\n\n/-%%\nLet $\\psi$ be a bumpfunction.\n\\begin{theorem}[SmoothExistence]\\label{SmoothExistence}\\lean{SmoothExistence}\\leanok\nThere exists a smooth (once differentiable would be enough), nonnegative ``bumpfunction'' $\\psi$,\n supported in $[1/2,2]$ with total mass one:\n$$\n\\int_0^\\infty \\psi(x)\\frac{dx}{x} = 1.\n$$\n\\end{theorem}\n%%-/\n\nattribute [- simp] one_div in\n\nlemma SmoothExistence : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n    \u03a8.support \u2286 Icc (1 / 2) 2 \u2227 \u222b x in Ici 0, \u03a8 x / x = 1 := by\n  suffices h : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n      \u03a8.support \u2286 Set.Icc (1 / 2) 2 \u2227 0 < \u222b x in Set.Ici 0, \u03a8 x / x by\n    rcases h with \u27e8\u03a8, h\u03a8, h\u03a8nonneg, h\u03a8supp, h\u03a8pos\u27e9\n    let c := (\u222b x in Ici 0, \u03a8 x / x)\n    use fun y \u21a6 \u03a8 y / c\n    refine \u27e8h\u03a8.div_const c, fun y \u21a6 div_nonneg (h\u03a8nonneg y) (le_of_lt h\u03a8pos), ?_, ?_\u27e9\n    \u00b7 rw [Function.support_div, Function.support_const (ne_of_lt h\u03a8pos).symm, inter_univ]\n      convert h\u03a8supp\n    \u00b7 simp only [div_right_comm _ c _, integral_div c, div_self <| ne_of_gt h\u03a8pos]\n\n  have := smooth_urysohn_support_Ioo (a := 1 / 2) (b := 1) (c := 3/2) (d := 2) (by linarith)\n    (by linarith)\n  rcases this with \u27e8\u03a8, h\u03a8ContDiff, _, h\u03a80, h\u03a81, h\u03a8Support\u27e9\n  use \u03a8, h\u03a8ContDiff\n  unfold indicator at h\u03a80 h\u03a81\n  simp only [mem_Icc, Pi.one_apply, Pi.le_def, mem_Ioo] at h\u03a80 h\u03a81\n  simp only [h\u03a8Support, subset_def, mem_Ioo, mem_Icc, and_imp]\n  split_ands\n  \u00b7 exact fun x \u21a6 le_trans (by simp [apply_ite]) (h\u03a80 x)\n  \u00b7 exact fun y hy hy' \u21a6 \u27e8by linarith, by linarith\u27e9\n  \u00b7 rw [integral_pos_iff_support_of_nonneg]\n    \u00b7 simp only [Function.support_div, measurableSet_Ici, Measure.restrict_apply', h\u03a8Support, Function.support_id]\n      have : (Ioo (1 / 2 : \u211d) 2 \u2229 (Iio 0 \u222a Ioi 0) \u2229 Ici 0) = Ioo (1 / 2) 2 := by\n        ext x\n        simp only [mem_inter_iff, mem_Ioo, mem_Ici, mem_Iio, mem_Ioi,\n          mem_union, not_lt, and_true, not_le]\n        constructor\n        \u00b7 exact fun h \u21a6 h.left.left\n        \u00b7 intro h\n          simp only [h, and_self, lt_or_lt_iff_ne, ne_eq, true_and]\n          constructor <;> linarith [h.left]\n      simp only [this, volume_Ioo, ENNReal.ofReal_pos, sub_pos, gt_iff_lt]\n      linarith\n    \u00b7 simp_rw [Pi.le_def, Pi.zero_apply]\n      intro y\n      by_cases h : y \u2208 Function.support \u03a8\n      . apply div_nonneg <| le_trans (by simp [apply_ite]) (h\u03a80 y)\n        rw [h\u03a8Support, mem_Ioo] at h\n        linarith [h.left]\n      . simp only [Function.mem_support, ne_eq, not_not] at h\n        simp [h]\n    \u00b7 have : (fun x \u21a6 \u03a8 x / x).support \u2286 Icc (1 / 2) 2 := by\n        rw [Function.support_div, h\u03a8Support]\n        apply subset_trans (by apply inter_subset_left) Ioo_subset_Icc_self\n      apply (integrableOn_iff_integrable_of_support_subset this).mp\n      apply ContinuousOn.integrableOn_compact isCompact_Icc\n      apply ContinuousOn.div h\u03a8ContDiff.continuous.continuousOn continuousOn_id ?_\n      simp only [mem_Icc, ne_eq, and_imp, id_eq]\n      intros\n      linarith\n/-%%\n\\begin{proof}\\leanok\n\\uses{smooth-ury}\nSame idea as Urysohn-type argument.\n\\end{proof}\n%%-/\n\nlemma mem_within_strip (\u03c3\u2081 \u03c3\u2082 : \u211d) :\n  {s : \u2102 | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} \u2208 \ud835\udcdf {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} := by simp\n\nlemma MellinOfPsi_aux {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {s : \u2102} (hs : s \u2260 0) :\n    \u222b (x : \u211d) in Ioi 0, (\u03a8 x) * (x : \u2102) ^ (s - 1) =\n    - (1 / s) * \u222b (x : \u211d) in Ioi 0, (deriv \u03a8 x) * (x : \u2102) ^ s := by\n  let g {s : \u2102} := fun (x : \u211d)  \u21a6 x ^ s / s\n  have gderiv {s : \u2102} (hs : s \u2260 0) {x: \u211d} (hx : x \u2208 Ioi 0) :\n      deriv g x = x ^ (s - 1) := by\n    have := HasDerivAt.cpow_const (c := s) (hasDerivAt_id (x : \u2102)) (Or.inl hx)\n    simp_rw [mul_one, id_eq] at this\n    rw [deriv_div_const, deriv.comp_ofReal (e := fun x \u21a6 x ^ s)]\n    \u00b7 rw [this.deriv, mul_div_right_comm, div_self hs, one_mul]\n    \u00b7 apply hasDerivAt_deriv_iff.mp\n      simp only [this.deriv, this]\n  calc\n    _ =  \u222b (x : \u211d) in Ioi 0, \u2191(\u03a8 x) * deriv (@g s) x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv (fun x \u21a6 \u2191(\u03a8 x)) x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * x ^ s / s := by simp only [mul_div, g]\n    _ = _ := ?_\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro _ hx\n    simp only [gderiv hs hx]\n  \u00b7 apply PartialIntegration_of_support_in_Icc (\u03a8 \u00b7) g\n      (a := 1 / 2) (b := 2) (by norm_num) (by norm_num)\n    \u00b7 simpa only [Function.support_subset_iff, ne_eq, ofReal_eq_zero]\n    \u00b7 exact (Differentiable.ofReal_comp_iff.mpr (diff\u03a8.differentiable (by norm_num))).differentiableOn\n    \u00b7 refine DifferentiableOn.div_const ?_ s\n      intro a ha\n      refine DifferentiableAt.differentiableWithinAt ?_\n      apply DifferentiableAt.comp_ofReal (e := fun x \u21a6 x ^ s)\n      apply DifferentiableAt.cpow differentiableAt_id' <| differentiableAt_const s\n      exact Or.inl ha\n    \u00b7 simp only [deriv.ofReal_comp']\n      apply Continuous.continuousOn\n      apply Continuous.comp (g := ofReal') continuous_ofReal <| diff\u03a8.continuous_deriv (by norm_num)\n    \u00b7 apply ContinuousOn.congr (f := fun (x : \u211d) \u21a6 (x : \u2102) ^ (s - 1)) ?_ fun x hx \u21a6 gderiv hs hx\n      refine ContinuousOn.cpow ?_ continuousOn_const (by simp)\n      exact Continuous.continuousOn (by continuity)\n  \u00b7 congr; funext; congr\n    apply (hasDerivAt_deriv_iff.mpr ?_).ofReal_comp.deriv\n    exact diff\u03a8.contDiffAt.differentiableAt (by norm_num)\n  \u00b7 simp only [neg_mul, neg_inj]\n    conv => lhs; rhs; intro; rw [\u2190 mul_one_div, mul_comm]\n    rw [integral_mul_left]\n\n/-%%\nThe $\\psi$ function has Mellin transform $\\mathcal{M}(\\psi)(s)$ which is entire and decays (at\nleast) like $1/|s|$.\n\\begin{theorem}[MellinOfPsi]\\label{MellinOfPsi}\\lean{MellinOfPsi}\\leanok\nThe Mellin transform of $\\psi$ is\n$$\\mathcal{M}(\\psi)(s) =  O\\left(\\frac{1}{|s|}\\right),$$\nas $|s|\\to\\infty$ with $\\sigma_1 \\le \\Re(s) \\le \\sigma_2$.\n\\end{theorem}\n\n[Of course it decays faster than any power of $|s|$, but it turns out that we will just need one\npower.]\n%%-/\nlemma MellinOfPsi {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {\u03c3\u2081 \u03c3\u2082 : \u211d} (\u03c3\u2081pos : 0 < \u03c3\u2081) :\n    (fun s \u21a6 \u2016\ud835\udcdc (\u03a8 \u00b7) s\u2016)\n    =O[Filter.principal {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082}]\n      fun s \u21a6 1 / \u2016s\u2016 := by\n  let f := fun (x : \u211d) \u21a6 \u2016deriv \u03a8 x\u2016\n  have cont : ContinuousOn f (Icc (1 / 2) 2) :=\n    (Continuous.comp (by continuity) <| diff\u03a8.continuous_deriv (by norm_num)).continuousOn\n  obtain \u27e8a, _, max\u27e9 := isCompact_Icc.exists_isMaxOn (f := f) (by norm_num) cont\n  rw [Asymptotics.isBigO_iff]\n  use f a * 2 ^ \u03c3\u2082 * (3 / 2)\n  filter_upwards [mem_within_strip \u03c3\u2081 \u03c3\u2082] with s hs\n  have s_ne_zero: s \u2260 0 := fun h \u21a6 by linarith [zero_re \u25b8 h \u25b8 hs.1]\n  simp only [MellinTransform, f, MellinOfPsi_aux diff\u03a8 supp\u03a8 s_ne_zero, norm_norm, norm_mul]\n  conv => rhs; rw [mul_comm]\n  gcongr; simp\n  calc\n    _ \u2264 \u222b (x : \u211d) in Ioi 0, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016 := ?_\n    _ = \u222b (x : \u211d) in Icc (1 / 2) 2, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016 := ?_\n    _ \u2264 \u2016\u222b (x : \u211d) in Icc (1 / 2) 2, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016\u2016 := le_abs_self _\n    _ \u2264 _ := ?_\n  \u00b7 simp_rw [norm_integral_le_integral_norm]\n  \u00b7 apply SetIntegral.integral_eq_integral_inter_of_support_subset_Icc\n    \u00b7 simp only [Function.support_abs, Function.support_mul, Function.support_ofReal]\n      apply subset_trans (by apply inter_subset_left) <| Function.support_deriv_subset_Icc supp\u03a8\n    \u00b7 exact (Icc_subset_Ioi_iff (by norm_num)).mpr (by norm_num)\n  \u00b7 have := intervalIntegral.norm_integral_le_of_norm_le_const' (C := f a * 2 ^ \u03c3\u2082)\n      (f := fun x \u21a6 f x * \u2016(x : \u2102) ^ s\u2016) (a := (1 / 2 : \u211d)) ( b := 2) (by norm_num) ?_\n    \u00b7 simp only [Real.norm_eq_abs, Complex.norm_eq_abs, abs_ofReal, map_mul] at this \u22a2\n      rwa [(by norm_num: |(2 : \u211d) - 1 / 2| = 3 / 2),\n          intervalIntegral.integral_of_le (by norm_num), \u2190 integral_Icc_eq_integral_Ioc] at this\n    \u00b7 intro x hx;\n      have f_bound := isMaxOn_iff.mp max x hx\n      have pow_bound : \u2016(x : \u2102) ^ s\u2016 \u2264 2 ^ \u03c3\u2082 := by\n        rw [Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos (by linarith [mem_Icc.mp hx])]\n        have xpos : 0 \u2264 x := by linarith [(mem_Icc.mp hx).1]\n        have h := rpow_le_rpow xpos (mem_Icc.mp hx).2 (by linarith : 0 \u2264 s.re)\n        exact le_trans h <| rpow_le_rpow_of_exponent_le (by norm_num) hs.2\n      convert mul_le_mul f_bound pow_bound (norm_nonneg _) ?_ using 1 <;> simp [f]\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform, SmoothExistence}\nIntegrate by parts:\n$$\n\\left|\\int_0^\\infty \\psi(x)x^s\\frac{dx}{x}\\right| =\n\\left|-\\int_0^\\infty \\psi'(x)\\frac{x^{s}}{s}dx\\right|\n$$\n$$\n\\le \\frac{1}{|s|} \\int_{1/2}^2|\\psi'(x)|x^{\\Re(s)}dx.\n$$\nSince $\\Re(s)$ is bounded, the right-hand side is bounded by a\nconstant times $1/|s|$.\n\\end{proof}\n%%-/\n\n/-%%\nWe can make a delta spike out of this bumpfunction, as follows.\n\\begin{definition}[DeltaSpike]\\label{DeltaSpike}\\lean{DeltaSpike}\\leanok\n\\uses{SmoothExistence}\nLet $\\psi$ be a bumpfunction supported in $[1/2,2]$. Then for any $\\epsilon>0$, we define the\ndelta spike $\\psi_\\epsilon$ to be the function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\psi_\\epsilon(x) = \\frac{1}{\\epsilon}\\psi\\left(x^{\\frac{1}{\\epsilon}}\\right).$$\n\\end{definition}\n%%-/\n\nnoncomputable def DeltaSpike (\u03a8 : \u211d \u2192 \u211d) (\u03b5 : \u211d) : \u211d \u2192 \u211d :=\n  fun x \u21a6 \u03a8 (x ^ (1 / \u03b5)) / \u03b5\n\n/-%%\nThis spike still has mass one:\n\\begin{lemma}[DeltaSpikeMass]\\label{DeltaSpikeMass}\\lean{DeltaSpikeMass}\\leanok\nFor any $\\epsilon>0$, we have\n$$\\int_0^\\infty \\psi_\\epsilon(x)\\frac{dx}{x} = 1.$$\n\\end{lemma}\n%%-/\n\nlemma DeltaSpikeMass {\u03a8 : \u211d \u2192 \u211d} (mass_one: \u222b x in Ioi 0, \u03a8 x / x = 1) {\u03b5 : \u211d}\n    (\u03b5pos : 0 < \u03b5) : \u222b x in Ioi 0, ((DeltaSpike \u03a8 \u03b5) x) / x = 1 :=\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, (|1/\u03b5| * x ^ (1 / \u03b5 - 1)) \u2022\n      ((fun z \u21a6 (\u03a8 z) / z) (x ^ (1 / \u03b5))) := by\n      apply set_integral_congr_ae measurableSet_Ioi\n      filter_upwards with x hx\n      simp only [mem_Ioi, smul_eq_mul, abs_of_pos (one_div_pos.mpr \u03b5pos)]\n      symm; calc\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / x ^ (1 / \u03b5)) * x ^ (1 / \u03b5 - 1) * (1 / \u03b5) := by ring\n        _ = _ := by rw [rpow_sub hx, rpow_one]\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / x ^ (1 / \u03b5) * x ^ (1 / \u03b5) / x) * (1/ \u03b5) := by ring\n        _ = _ := by rw [div_mul_cancel\u2080 _ (ne_of_gt (rpow_pos_of_pos hx (1/\u03b5)))]\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / \u03b5 / x) := by ring\n    _ = 1 := by\n      rw [integral_comp_rpow_Ioi (fun z \u21a6 (\u03a8 z) / z), \u2190 mass_one]\n      simp only [ne_eq, div_eq_zero_iff, one_ne_zero, \u03b5pos.ne', or_self, not_false_eq_true]\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{DeltaSpike}\nSubstitute $y=x^{1/\\epsilon}$, and use the fact that $\\psi$ has mass one, and that $dx/x$ is Haar\nmeasure.\n\\end{proof}\n%%-/\n\nlemma DeltaSpikeSupport_aux {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2):\n    (fun x \u21a6 if x < 0 then 0 else DeltaSpike \u03a8 \u03b5 x).support \u2286 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) := by\n  unfold DeltaSpike\n  simp only [one_div, Function.support_subset_iff, ne_eq, ite_eq_left_iff, not_lt, div_eq_zero_iff,\n    not_forall, exists_prop, mem_Icc, and_imp]\n  intro x hx h; push_neg at h\n  have := supp\u03a8 <| Function.mem_support.mpr h.1\n  simp only [one_div, mem_Icc] at this\n  have hl := (le_rpow_inv_iff_of_pos (by norm_num) hx \u03b5pos).mp this.1\n  rw [inv_rpow (by norm_num) \u03b5, \u2190 rpow_neg (by norm_num)] at hl\n  refine \u27e8hl, (rpow_inv_le_iff_of_pos ?_ (by norm_num) \u03b5pos).mp this.2\u27e9\n  linarith [(by apply rpow_nonneg (by norm_num) : 0 \u2264 (2 : \u211d) ^ (-\u03b5))]\n\nlemma DeltaSpikeSupport' {\u03a8 : \u211d \u2192 \u211d} {\u03b5 x : \u211d} (\u03b5pos : 0 < \u03b5) (xnonneg : 0 \u2264 x)\n    (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2) :\n    DeltaSpike \u03a8 \u03b5 x \u2260 0 \u2192 x \u2208 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) := by\n  intro h\n  have : (fun x \u21a6 if x < 0 then 0 else DeltaSpike \u03a8 \u03b5 x) x = DeltaSpike \u03a8 \u03b5 x := by simp [xnonneg]\n  rw [\u2190 this] at h\n  exact (Function.support_subset_iff.mp <| DeltaSpikeSupport_aux \u03b5pos supp\u03a8) _ h\n\nlemma DeltaSpikeSupport {\u03a8 : \u211d \u2192 \u211d} {\u03b5 x : \u211d} (\u03b5pos : 0 < \u03b5) (xnonneg : 0 \u2264 x)\n    (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2) :\n    x \u2209 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) \u2192 DeltaSpike \u03a8 \u03b5 x = 0 := by\n  contrapose!; exact DeltaSpikeSupport' \u03b5pos xnonneg supp\u03a8\n\nlemma DeltaSpikeContinuous {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (diff\u03a8 : ContDiff \u211d 1 \u03a8) :\n    Continuous (fun x \u21a6 DeltaSpike \u03a8 \u03b5 x) := by\n  apply (Continuous.comp (g := \u03a8) diff\u03a8.continuous _).div_const\n  exact Continuous.rpow_const continuous_id fun _ \u21a6 Or.inr <| div_nonneg (by norm_num) \u03b5pos.le\n\nlemma DeltaSpikeOfRealContinuous {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (diff\u03a8 : ContDiff \u211d 1 \u03a8) :\n    Continuous (fun x \u21a6 (DeltaSpike \u03a8 \u03b5 x : \u2102)) :=\n  Continuous.comp continuous_ofReal <| DeltaSpikeContinuous \u03b5pos diff\u03a8\n\n/-%%\nThe Mellin transform of the delta spike is easy to compute.\n\\begin{theorem}[MellinOfDeltaSpike]\\label{MellinOfDeltaSpike}\\lean{MellinOfDeltaSpike}\\leanok\nFor any $\\epsilon>0$, the Mellin transform of $\\psi_\\epsilon$ is\n$$\\mathcal{M}(\\psi_\\epsilon)(s) = \\mathcal{M}(\\psi)\\left(\\epsilon s\\right).$$\n\\end{theorem}\n%%-/\ntheorem MellinOfDeltaSpike (\u03a8 : \u211d \u2192 \u211d) {\u03b5 : \u211d} (\u03b5pos : \u03b5 > 0) (s : \u2102) :\n    \ud835\udcdc ((DeltaSpike \u03a8 \u03b5) \u00b7) s = \ud835\udcdc (\u03a8 \u00b7) (\u03b5 * s) := by\n  unfold MellinTransform DeltaSpike\n  rw [\u2190 integral_comp_rpow_Ioi (fun z \u21a6 ((\u03a8 z) : \u2102) * (z : \u2102) ^ ((\u03b5 : \u2102) * s - 1))\n    (one_div_ne_zero (ne_of_gt \u03b5pos))]\n  apply set_integral_congr_ae measurableSet_Ioi\n  filter_upwards with x hx\n\n  -- Simple algebra, would be nice if some tactic could handle this\n  have log_x_real: (Complex.log (x : \u2102)).im = 0 := by\n    rw [\u2190 ofReal_log, ofReal_im]\n    exact LT.lt.le hx\n  rw [div_eq_mul_inv, ofReal_mul, abs_of_pos (one_div_pos.mpr \u03b5pos)]\n  simp only [real_smul, ofReal_mul, ofReal_div, ofReal_one, Complex.ofReal_rpow hx]\n  rw [\u2190 Complex.cpow_mul _ ?_ ?_, mul_sub]\n  \u00b7 simp only [\u2190 mul_assoc, ofReal_sub, ofReal_div, ofReal_one, mul_one, ofReal_inv]\n    symm\n    \u00b7 rw [one_div_mul_cancel (by exact slitPlane_ne_zero (Or.inl \u03b5pos)), mul_comm (1 / (\u03b5:\u2102)),\n          mul_comm, \u2190 mul_assoc, \u2190 mul_assoc]\n      rw [\u2190 Complex.cpow_add _ _ (by exact slitPlane_ne_zero (Or.inl hx))]; ring_nf\n  \u00b7 simp [im_mul_ofReal, log_x_real, zero_mul, pi_pos]\n  \u00b7 simp [im_mul_ofReal, log_x_real, zero_mul, pi_nonneg]\n/-%%\n\\begin{proof}\\leanok\n\\uses{DeltaSpike, MellinTransform}\nSubstitute $y=x^{1/\\epsilon}$, use Haar measure; direct calculation.\n\\end{proof}\n%%-/\n\n/-%%\nIn particular, for $s=1$, we have that the Mellin transform of $\\psi_\\epsilon$ is $1+O(\\epsilon)$.\n\\begin{corollary}[MellinOfDeltaSpikeAt1]\\label{MellinOfDeltaSpikeAt1}\\lean{MellinOfDeltaSpikeAt1}\n\\leanok\nFor any $\\epsilon>0$, we have\n$$\\mathcal{M}(\\psi_\\epsilon)(1) =\n\\mathcal{M}(\\psi)(\\epsilon).$$\n\\end{corollary}\n%%-/\n\nlemma MellinOfDeltaSpikeAt1 (\u03a8 : \u211d \u2192 \u211d) {\u03b5 : \u211d} (\u03b5pos : \u03b5 > 0) :\n    \ud835\udcdc ((DeltaSpike \u03a8 \u03b5) \u00b7) 1 = \ud835\udcdc (\u03a8 \u00b7) \u03b5 := by\n  convert MellinOfDeltaSpike \u03a8 \u03b5pos 1; simp [mul_one]\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinOfDeltaSpike, DeltaSpikeMass}\nThis is immediate from the above theorem.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[MellinOfDeltaSpikeAt1_asymp]\\label{MellinOfDeltaSpikeAt1_asymp}\n\\lean{MellinOfDeltaSpikeAt1_asymp}\\leanok\nAs $\\epsilon\\to 0$, we have\n$$\\mathcal{M}(\\psi_\\epsilon)(1) = 1+O(\\epsilon).$$\n\\end{lemma}\n%%-/\nlemma MellinOfDeltaSpikeAt1_asymp {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    (mass_one : \u222b x in Set.Ioi 0, \u03a8 x / x = 1) :\n    (fun (\u03b5 : \u211d) \u21a6 (\ud835\udcdc (\u03a8 \u00b7) \u03b5) - 1) =O[\ud835\udcdd[>]0] id := by\n  have diff : DifferentiableWithinAt \u211d (fun (\u03b5 : \u211d) \u21a6 \ud835\udcdc (\u03a8 \u00b7) \u03b5 - 1) (Ioi 0) 0 := by\n    apply DifferentiableAt.differentiableWithinAt\n    simp only [differentiableAt_sub_const_iff, MellinTransform_eq]\n    refine DifferentiableAt.comp_ofReal ?_\n    refine mellin_differentiableAt_of_isBigO_rpow (a := 1) (b := -1) ?_ ?_ (by simp) ?_ (by simp)\n    \u00b7 apply ContinuousOn.locallyIntegrableOn (Continuous.continuousOn ?_) (by simp)\n      have := diff\u03a8.continuous; continuity\n    \u00b7 apply Asymptotics.IsBigO.trans_le (g' := fun _ \u21a6 (0 : \u211d)) ?_ (by simp)\n      apply BigO_zero_atTop_of_support_in_Icc (a := 1 / 2) (b := 2)\n      rwa [\u03a8.support_ofReal]\n    \u00b7 apply Asymptotics.IsBigO.trans_le (g' := fun _ \u21a6 (0 : \u211d)) ?_ (by simp)\n      apply BigO_zero_atZero_of_support_in_Icc (a := 1 / 2) (b := 2) (ha := (by norm_num))\n      rwa [\u03a8.support_ofReal]\n  have := ofReal_zero \u25b8 diff.isBigO_sub\n  simp only [sub_sub_sub_cancel_right, sub_zero] at this\n  convert this\n  simp only [MellinTransform, zero_sub, sub_right_inj, cpow_neg_one, \u2190 div_eq_mul_inv, \u2190 ofReal_div]\n  rw [\u2190 ofReal_one, \u2190 mass_one]; convert integral_ofReal.symm\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform,MellinOfDeltaSpikeAt1,SmoothExistence}\nBy Lemma \\ref{MellinOfDeltaSpikeAt1},\n$$\n  \\mathcal M(\\psi_\\epsilon)(1)=\\mathcal M(\\psi)(\\epsilon)\n$$\nwhich by Definition \\ref{MellinTransform} is\n$$\n  \\mathcal M(\\psi)(\\epsilon)=\\int_0^\\infty\\psi(x)x^{\\epsilon-1}dx\n  .\n$$\nSince $\\psi(x) x^{\\epsilon-1}$ is integrable (because $\\psi$ is continuous and compactly supported),\n$$\n  \\mathcal M(\\psi)(\\epsilon)-\\int_0^\\infty\\psi(x)\\frac{dx}x=\\int_0^\\infty\\psi(x)(x^{\\epsilon-1}-x^{-1})dx\n  .\n$$\nBy Taylor's theorem,\n$$\n  x^{\\epsilon-1}-x^{-1}=O(\\epsilon)\n$$\nso, since $\\psi$ is absolutely integrable,\n$$\n  \\mathcal M(\\psi)(\\epsilon)-\\int_0^\\infty\\psi(x)\\frac{dx}x=O(\\epsilon)\n  .\n$$\nWe conclude the proof using Theorem \\ref{SmoothExistence}.\n\\end{proof}\n%%-/\n\n/-%%\nLet $1_{(0,1]}$ be the function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$1_{(0,1]}(x) = \\begin{cases}\n1 & \\text{ if }x\\leq 1\\\\\n0 & \\text{ if }x>1\n\\end{cases}.$$\nThis has Mellin transform\n\\begin{theorem}[MellinOf1]\\label{MellinOf1}\\lean{MellinOf1}\\leanok\nThe Mellin transform of $1_{(0,1]}$ is\n$$\\mathcal{M}(1_{(0,1]})(s) = \\frac{1}{s}.$$\n\\end{theorem}\n[Note: this already exists in mathlib]\n%%-/\nlemma MellinOf1 (s : \u2102) (h : s.re > 0) : \ud835\udcdc ((fun x \u21a6 if 0 < x \u2227 x \u2264 1 then 1 else 0)) s = 1 / s := by\n  convert (hasMellin_one_Ioc h).right using 1\n  apply set_integral_congr_ae measurableSet_Ioi\n  filter_upwards with _ _; rw [smul_eq_mul, mul_comm]; congr\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform}\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-%%\nWhat will be essential for us is properties of the smooth version of $1_{(0,1]}$, obtained as the\n Mellin convolution of $1_{(0,1]}$ with $\\psi_\\epsilon$.\n\\begin{definition}[Smooth1]\\label{Smooth1}\\lean{Smooth1}\n\\uses{MellinOf1, MellinConvolution}\\leanok\nLet $\\epsilon>0$. Then we define the smooth function $\\widetilde{1_{\\epsilon}}$ from\n$\\mathbb{R}_{>0}$ to $\\mathbb{C}$ by\n$$\\widetilde{1_{\\epsilon}} = 1_{(0,1]}\\ast\\psi_\\epsilon.$$\n\\end{definition}\n%%-/\nnoncomputable def Smooth1 (\u03a8 : \u211d \u2192 \u211d) (\u03b5 : \u211d) : \u211d \u2192 \u211d :=\n  MellinConvolution (fun x \u21a6 if 0 < x \u2227 x \u2264 1 then 1 else 0) (DeltaSpike \u03a8 \u03b5)\n\n/-%%\n\\begin{lemma}[Smooth1Properties_estimate]\\label{Smooth1Properties_estimate}\n\\lean{Smooth1Properties_estimate}\\leanok\nFor $\\epsilon>0$,\n$$\n  \\log2>\\frac{1-2^{-\\epsilon}}\\epsilon\n$$\n\\end{lemma}\n%%-/\n\nlemma Smooth1Properties_estimate {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n    (1 - 2 ^ (-\u03b5)) / \u03b5 < Real.log 2 := by\n  apply (div_lt_iff' \u03b5pos).mpr\n  have : 1 - 1 / (2 : \u211d) ^ \u03b5 = ((2 : \u211d) ^ \u03b5 - 1) / (2 : \u211d) ^ \u03b5 := by\n    rw [sub_div, div_self (by positivity)]\n  rw [\u2190 Real.log_rpow (by norm_num), rpow_neg (by norm_num), inv_eq_one_div (2 ^ \u03b5), this]\n  set c := (2 : \u211d) ^ \u03b5\n  have hc : 1 < c := by\n    rw [\u2190 rpow_zero (2 : \u211d)]\n    apply Real.rpow_lt_rpow_of_exponent_lt (by norm_num) \u03b5pos\n  apply (div_lt_iff' (by positivity)).mpr <| lt_sub_iff_add_lt'.mp ?_\n  let f := (fun x \u21a6 x * Real.log x - x)\n  rw [(by simp [f] : -1 = f 1), (by simp : c * Real.log c - c = f c)]\n  have mono: StrictMonoOn f <| Ici 1 := by\n    refine strictMonoOn_of_deriv_pos (convex_Ici _) ?_ ?_\n    \u00b7 apply ContinuousOn.sub (ContinuousOn.mul continuousOn_id ?_) continuousOn_id\n      apply ContinuousOn.log continuousOn_id\n      intro x hx; simp only [mem_Ici] at hx; simp only [id_eq, ne_eq]; linarith\n    \u00b7 intro x hx; simp only [nonempty_Iio, interior_Ici', mem_Ioi] at hx\n      funext; dsimp only [f]\n      rw [deriv_sub, deriv_mul, deriv_log, deriv_id'', one_mul, mul_inv_cancel]; simp\n      \u00b7 exact log_pos hx\n      \u00b7 linarith\n      \u00b7 simp only [differentiableAt_id']\n      \u00b7 simp only [differentiableAt_log_iff, ne_eq]; linarith\n      \u00b7 apply DifferentiableAt.mul differentiableAt_id' <| DifferentiableAt.log differentiableAt_id' ?_\n        linarith\n      \u00b7 simp only [differentiableAt_id']\n  exact mono (by rw [mem_Ici]) (mem_Ici.mpr <| le_of_lt hc) hc\n/-%%\n\\begin{proof}\\leanok\nLet $c:=2^\\epsilon > 1$, in terms of which we wish to prove\n$$\n  -1 < c \\log c - c .\n$$\nLetting $f(x):=x\\log x - x$, we can rewrite this as $f(1) < f(c)$.\nSince\n$$\n  \\frac {d}{dx}f(x) = \\log x > 0 ,\n$$\n$f$ is monotone increasing on [1, \\infty), and we are done.\n\\end{proof}\n%%-/\n\n\n/-%%\nIn particular, we have the following two properties.\n\\begin{lemma}[Smooth1Properties_below]\\label{Smooth1Properties_below}\n\\lean{Smooth1Properties_below}\\leanok\nFix $\\epsilon>0$. There is an absolute constant $c>0$ so that:\nIf $0 < x \\leq (1-c\\epsilon)$, then\n$$\\widetilde{1_{\\epsilon}}(x) = 1.$$\n\\end{lemma}\n%%-/\n\nlemma Smooth1Properties_below_aux {x \u03b5 : \u211d} (hx : x \u2264 1 - (2:\u211d).log * \u03b5) (\u03b5pos: 0 < \u03b5) :\n    x < 2 ^ (-\u03b5) := by\n  calc\n    x \u2264 1 - (2 : \u211d).log * \u03b5 := hx\n    _ < 2 ^ (-\u03b5) := ?_\n  rw [sub_lt_iff_lt_add, add_comm, \u2190 sub_lt_iff_lt_add]\n  exact (div_lt_iff \u03b5pos).mp <| Smooth1Properties_estimate \u03b5pos\n\nlemma Smooth1Properties_below {\u03a8 : \u211d \u2192 \u211d} (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2)\n    (\u03b5 : \u211d) (\u03b5pos: 0 < \u03b5) (mass_one : \u222b x in Ioi 0, \u03a8 x / x = 1) :\n    \u2203 (c : \u211d), 0 < c \u2227 \u2200 (x : \u211d), 0 < x \u2192 x \u2264 1 - c * \u03b5 \u2192 Smooth1 \u03a8 \u03b5 x = 1 := by\n  set c := (2 : \u211d).log; use c\n  constructor; exact log_pos (by norm_num)\n  intro x xpos hx\n  have hx2 := Smooth1Properties_below_aux hx \u03b5pos\n  rewrite [\u2190 DeltaSpikeMass mass_one \u03b5pos]\n  unfold Smooth1 MellinConvolution\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, indicator (Ioc 0 1) (fun y \u21a6 DeltaSpike \u03a8 \u03b5 (x / y) / \u2191y) y := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, DeltaSpike \u03a8 \u03b5 (x / y) / y := ?_\n    _ = _ := integral_comp_div_I0i_haar (fun y \u21a6 DeltaSpike \u03a8 \u03b5 y) xpos\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro y hy\n    by_cases h : y \u2264 1 <;> simp [indicator, mem_Ioi.mp hy, h]\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro y hy\n    simp only [indicator_apply_eq_self, mem_Ioc, not_and, not_le, div_eq_zero_iff]\n    intro hy2; replace hy2 := hy2 <| mem_Ioi.mp hy\n    by_cases h : y = 0; right; exact h; left\n    apply DeltaSpikeSupport \u03b5pos ?_ supp\u03a8\n    \u00b7 simp only [mem_Icc, not_and, not_le]; intro\n      linarith [(by apply (div_lt_iff (by linarith)).mpr; nlinarith : x / y < 2 ^ (-\u03b5))]\n    \u00b7 rw [le_div_iff (by linarith), zero_mul]; exact xpos.le\n/-%%\n\\begin{proof}\\leanok\n\\uses{Smooth1, MellinConvolution,DeltaSpikeMass, Smooth1Properties_estimate}\nOpening the definition, we have that the Mellin convolution of $1_{(0,1]}$ with $\\psi_\\epsilon$ is\n$$\n\\int_0^\\infty 1_{(0,1]}(y)\\psi_\\epsilon(x/y)\\frac{dy}{y}\n=\n\\int_0^1 \\psi_\\epsilon(x/y)\\frac{dy}{y}.\n$$\nThe support of $\\psi_\\epsilon$ is contained in $[1/2^\\epsilon,2^\\epsilon]$, so it suffices to consider\n$y \\in [1/2^\\epsilon x,2^\\epsilon x]$ for nonzero contributions. If $x < 2^{-\\epsilon}$, then the integral is the same as that over $(0,\\infty)$:\n$$\n\\int_0^1 \\psi_\\epsilon(x/y)\\frac{dy}{y}\n=\n\\int_0^\\infty \\psi_\\epsilon(x/y)\\frac{dy}{y},\n$$\nin which we change variables to $z=x/y$ (using $x>0$):\n$$\n\\int_0^\\infty \\psi_\\epsilon(x/y)\\frac{dy}{y}\n=\n\\int_0^\\infty \\psi_\\epsilon(z)\\frac{dz}{z},\n$$\nwhich is equal to one by Lemma \\ref{DeltaSpikeMass}.\nWe then choose\n$$\n  c:=\\log 2,\n$$\nwhich satisfies\n$$\n  c > \\frac{1-2^{-\\epsilon}}\\epsilon\n$$\nby Lemma \\ref{Smooth1Properties_estimate}, so\n$$\n  1-c\\epsilon < 2^{-\\epsilon}.\n$$\n\\end{proof}\n%%-/\n\nlemma Smooth1Properties_above_aux {x \u03b5 : \u211d} (hx : 1 + (2 * (2:\u211d).log) * \u03b5 \u2264 x) (h\u03b5 : \u03b5 \u2208 Ioo 0 1) :\n    2 ^ \u03b5 < x := by\n  calc\n    x \u2265 1 + (2 * (2 : \u211d).log) * \u03b5 := hx\n    _ > 2 ^ \u03b5 := ?_\n  refine lt_add_of_sub_left_lt <| (div_lt_iff h\u03b5.1).mp ?_\n  calc\n    2 * (2 : \u211d).log > 2 * (1 - 2 ^ (-\u03b5)) / \u03b5 := ?_\n    _ > 2 ^ \u03b5 * (1 - 2 ^ (-\u03b5)) / \u03b5 := ?_\n    _ = (2 ^ \u03b5 - 1) / \u03b5 := ?_\n  \u00b7 have := (mul_lt_mul_left (a := 2) (by norm_num)).mpr <| Smooth1Properties_estimate h\u03b5.1\n    field_simp at this; simp [this]\n  \u00b7 have : (2 : \u211d) ^ \u03b5 < 2 := by\n      nth_rewrite 1 [\u2190 pow_one 2]\n      convert rpow_lt_rpow_of_exponent_lt (x := 2) (by norm_num) h\u03b5.2 <;> norm_num\n    have pos: 0 < (1 - 2 ^ (-\u03b5)) / \u03b5 := by\n      refine div_pos ?_ h\u03b5.1\n      rw [sub_pos, \u2190 pow_zero 2]\n      convert rpow_lt_rpow_of_exponent_lt (x := 2) (by norm_num) (neg_lt_zero.mpr h\u03b5.1); norm_num\n    have := (mul_lt_mul_right pos).mpr this\n    ring_nf at this \u22a2\n    exact this\n  \u00b7 have : (2 : \u211d) ^ \u03b5 * (2 : \u211d) ^ (-\u03b5) = (2 : \u211d) ^ (\u03b5 - \u03b5) := by\n      rw [\u2190 rpow_add (by norm_num), add_neg_self, sub_self]\n    conv => lhs; lhs; ring_nf; rhs; simp [this]\n\nlemma Smooth1Properties_above_aux2 {x y \u03b5 : \u211d} (h\u03b5 : \u03b5 \u2208 Ioo 0 1) (hy : y \u2208 Ioc 0 1)\n  (hx2 : 2 ^ \u03b5 < x) :\n    2 < (x / y) ^ (1 / \u03b5) := by\n  obtain \u27e8\u03b5pos, \u03b51\u27e9 := h\u03b5\n  obtain \u27e8ypos, y1\u27e9 := hy\n  calc\n    _ > (2 ^ \u03b5 / y) ^ (1 / \u03b5) := ?_\n    _ = 2 / y ^ (1 / \u03b5) := ?_\n    _ \u2265 2 / y := ?_\n    _ \u2265 2 := ?_\n  \u00b7 rw [gt_iff_lt, div_rpow, div_rpow, lt_div_iff, mul_comm_div, div_self, mul_one]\n    <;> try positivity\n    \u00b7 exact rpow_lt_rpow (by positivity) hx2 (by positivity)\n    \u00b7 exact LT.lt.le <| lt_trans (by positivity) hx2\n  \u00b7 rw [div_rpow, \u2190 rpow_mul, mul_div_cancel\u2080 1 <| ne_of_gt \u03b5pos, rpow_one] <;> positivity\n  \u00b7 have : y ^ (1 / \u03b5) \u2264 y := by\n      nth_rewrite 2 [\u2190 rpow_one y]\n      exact rpow_le_rpow_of_exponent_ge ypos y1 (by linarith [one_lt_one_div \u03b5pos \u03b51])\n    have pos : 0 < y ^ (1 / \u03b5) := by apply rpow_pos_of_pos <| ypos\n    rw [ge_iff_le, div_le_iff, div_mul_eq_mul_div, le_div_iff', mul_comm] <;> try linarith\n  \u00b7 rw [ge_iff_le, le_div_iff <| ypos]; exact (mul_le_iff_le_one_right zero_lt_two).mpr y1\n/-%%\n\\begin{lemma}[Smooth1Properties_above]\\label{Smooth1Properties_above}\n\\lean{Smooth1Properties_above}\\leanok\nFix $0<\\epsilon<1$. There is an absolute constant $c>0$ so that:\nif $x\\geq (1+c\\epsilon)$, then\n$$\\widetilde{1_{\\epsilon}}(x) = 0.$$\n\\end{lemma}\n%%-/\nlemma Smooth1Properties_above {\u03a8 : \u211d \u2192 \u211d} (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2)\n    {\u03b5 : \u211d} (h\u03b5 : \u03b5 \u2208 Ioo 0 1) :\n    \u2203 (c : \u211d), 0 < c \u2227 \u2200 (x : \u211d), 1 + c * \u03b5 \u2264 x \u2192 Smooth1 \u03a8 \u03b5 x = 0 := by\n  set c := 2 * Real.log 2; use c\n  constructor\n  \u00b7 simp only [c, zero_lt_two, mul_pos_iff_of_pos_left]; exact log_pos (by norm_num)\n  intro x hx\n  have hx2 := Smooth1Properties_above_aux hx h\u03b5\n  unfold Smooth1 MellinConvolution\n  simp only [ite_mul, one_mul, zero_mul, RCLike.ofReal_real_eq_id, id_eq]\n  apply set_integral_eq_zero_of_forall_eq_zero\n  intro y hy\n  have ypos := mem_Ioi.mp hy\n  by_cases y1 : y \u2264 1; swap; simp [ypos, y1]\n  simp only [mem_Ioi.mp hy, y1, and_self, \u2193reduceIte, div_eq_zero_iff]; left\n  apply DeltaSpikeSupport h\u03b5.1 ?_ supp\u03a8\n  simp only [mem_Icc, not_and, not_le]\n  swap; suffices h : 2 ^ \u03b5 < x / y by\n    linarith [(by apply rpow_pos_of_pos (by norm_num) : 0 < (2 : \u211d) ^ \u03b5)]\n  all_goals\n  try intro\n  have : x / y = ((x / y) ^ (1 / \u03b5)) ^ \u03b5 := by\n    rw [\u2190 rpow_mul]\n    simp only [one_div, inv_mul_cancel (ne_of_gt h\u03b5.1), rpow_one]\n    apply div_nonneg_iff.mpr; left;\n    exact \u27e8(le_trans (rpow_pos_of_pos (by norm_num) \u03b5).le) hx2.le, ypos.le\u27e9\n  rw [this]\n  refine rpow_lt_rpow (by norm_num) ?_ h\u03b5.1\n  exact Smooth1Properties_above_aux2 h\u03b5 \u27e8ypos, y1\u27e9 hx2\n/-%%\n\\begin{proof}\\leanok\n\\uses{Smooth1, MellinConvolution, Smooth1Properties_estimate}\nAgain the Mellin convolution is\n$$\\int_0^1 \\psi_\\epsilon(x/y)\\frac{dy}{y},$$\nbut now if $x > 2^\\epsilon$, then the support of $\\psi_\\epsilon$ is disjoint\nfrom the region of integration, and hence the integral is zero.\nWe choose\n$$\n  c:=2\\log 2\n  .\n$$\nBy Lemma \\ref{Smooth1Properties_estimate},\n$$\n  c > 2\\frac{1-2^{-\\epsilon}}\\epsilon > 2^\\epsilon\\frac{1-2^{-\\epsilon}}\\epsilon\n  =\n  \\frac{2^\\epsilon-1}\\epsilon,\n$$\nso\n$$\n  1+c\\epsilon > 2^\\epsilon.\n$$\n\\end{proof}\n%%-/\n\nlemma DeltaSpikeNonNeg_of_NonNeg {\u03a8 : \u211d \u2192 \u211d} (\u03a8nonneg : \u2200 x > 0, 0 \u2264 \u03a8 x)\n     {x \u03b5 : \u211d} (xpos : 0 < x) (\u03b5pos : 0 < \u03b5) :\n    0 \u2264 DeltaSpike \u03a8 \u03b5 x := by\n  dsimp [DeltaSpike]\n  have : 0 < x ^ (1 / \u03b5) := by positivity\n  have : 0 \u2264 \u03a8 (x ^ (1 / \u03b5)) := \u03a8nonneg _ this\n  positivity\n\nlemma MellinConvNonNeg_of_NonNeg {f g : \u211d \u2192 \u211d} (f_nonneg : \u2200 x > 0, 0 \u2264 f x)\n    (g_nonneg : \u2200 x > 0, 0 \u2264 g x) {x : \u211d} (xpos : 0 < x) :\n    0 \u2264 MellinConvolution f g x := by\n  dsimp [MellinConvolution]\n  apply MeasureTheory.set_integral_nonneg\n  \u00b7 exact measurableSet_Ioi\n  \u00b7 intro y ypos; simp only [mem_Ioi] at ypos\n    have : 0 \u2264 f y := f_nonneg _ ypos\n    have : 0 < x / y := by positivity\n    have : 0 \u2264 g (x / y) := g_nonneg _ this\n    positivity\n\n/-%%\n\\begin{lemma}[Smooth1Nonneg]\\label{Smooth1Nonneg}\\lean{Smooth1Nonneg}\\leanok\nIf $\\psi$ is nonnegative, then $\\widetilde{1_{\\epsilon}}(x)$ is nonnegative.\n\\end{lemma}\n%%-/\nlemma Smooth1Nonneg {\u03a8 : \u211d \u2192 \u211d} (\u03a8nonneg : \u2200 x > 0, 0 \u2264 \u03a8 x) {\u03b5 x : \u211d} (xpos : 0 < x)\n    (\u03b5pos : 0 < \u03b5) : 0 \u2264 Smooth1 \u03a8 \u03b5 x := by\n  dsimp [Smooth1]\n  apply MellinConvNonNeg_of_NonNeg ?_ ?_ xpos\n  \u00b7 intro y hy; by_cases h : y \u2264 1 <;> simp [h, hy]\n  \u00b7 intro y ypos; exact DeltaSpikeNonNeg_of_NonNeg \u03a8nonneg ypos \u03b5pos\n/-%%\n\\begin{proof}\\uses{Smooth1, MellinConvolution, DeltaSpike}\\leanok\nBy Definitions \\ref{Smooth1}, \\ref{MellinConvolution} and \\ref{DeltaSpike}\n$$\n  \\widetilde{1_\\epsilon}(x)=\\int_0^\\infty 1_{(0,1]}(y)\\frac1\\epsilon\\psi((x/y)^{\\frac1\\epsilon}) \\frac{dy}y\n$$\nand all the factors in the integrand are nonnegative.\n\\end{proof}\n%%-/\n\n", "theoremStatement": "lemma Smooth1LeOne_aux {x \u03b5 : \u211d} {\u03a8 : \u211d \u2192 \u211d} (xpos : 0 < x) (\u03b5pos : 0 < \u03b5)\n    (mass_one : \u222b x in Ioi 0, \u03a8 x / x = 1) :\n    \u222b (y : \u211d) in Ioi 0, \u03a8 ((x / y) ^ (1 / \u03b5)) / \u03b5 / y = 1 ", "theoremName": "Smooth1LeOne_aux", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "d179e6b9f714148e89c95c913f776491d9678612", "date": "2024-04-21"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/MellinCalculus.lean", "module": "PrimeNumberTheoremAnd.MellinCalculus", "jsonFile": "PrimeNumberTheoremAnd.MellinCalculus.jsonl", "positionMetadata": {"lineInFile": 1398, "tokenPositionInFile": 58046, "theoremPositionInFile": 79}, 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"Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n    calc\n      _ = \u222b (y : \u211d) in Ioi 0, (\u03a8 (y ^ (1 / \u03b5)) / \u03b5) / y := ?_\n      _ = \u222b (y : \u211d) in Ioi 0, \u03a8 y / y := ?_\n      _ = 1 := mass_one\n    \u00b7 have := integral_comp_div_I0i_haar (fun y \u21a6 \u03a8 ((x / y) ^ (1 / \u03b5)) / \u03b5) xpos\n      convert this.symm using 1\n      congr; funext y; congr; field_simp [mul_comm]\n    \u00b7 have := integral_comp_rpow_I0i_haar_real (fun y \u21a6 \u03a8 y) (one_div_ne_zero \u03b5pos.ne')\n      field_simp [ \u2190 this, abs_of_pos <| one_div_pos.mpr \u03b5pos]", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 461}}
+{"srcContext": "import PrimeNumberTheoremAnd.Wiener\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.NumberTheory.PrimeCounting\nimport Mathlib.Analysis.Asymptotics.AsymptoticEquivalent\n\nopen ArithmeticFunction hiding log\nopen Nat hiding log\nopen Finset\nopen BigOperators Filter Real Classical Asymptotics MeasureTheory\n\n/-%%\n\\begin{lemma}\\label{range-eq-range}\\lean{finsum_range_eq_sum_range, finsum_range_eq_sum_range'}\\leanok For any arithmetic function $f$ and real number $x$, one has\n$$ \\sum_{n \\leq x} f(n) = \\sum_{n \\leq \u230ax\u230b_+} f(n)$$\nand\n$$ \\sum_{n < x} f(n) = \\sum_{n < \u2308x\u2309_+} f(n).$$\n\\end{lemma}\n%%-/\n", "theoremStatement": "lemma finsum_range_eq_sum_range {R: Type*} [AddCommMonoid R] {f : ArithmeticFunction R} (x : \u211d) :\n    \u2211\u1da0 (n : \u2115) (_: n < x), f n = \u2211 n in range \u2308x\u2309\u208a, f n ", "theoremName": 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"Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", 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"Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.ArithmeticFunction"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply finsum_cond_eq_sum_of_cond_iff f\n  intros\n  simp only [mem_range]\n  exact Iff.symm Nat.lt_ceil", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 108}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\n", "theoremStatement": "lemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 ", "theoremName": "sum_range_succ", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "d640445b6a36605c2ae7327c033a4cd007da8dcb", "date": "2024-03-25"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1187, "tokenPositionInFile": 59506, "theoremPositionInFile": 111}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 86, 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"Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 139}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\n", "theoremStatement": "lemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) ", "theoremName": 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"Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done", "proofType": "tactic", "proofLengthLines": 51, "proofLengthTokens": 1979}}
+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\ntheorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) := by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz\n\ntheorem primesBetween_le_siftedSum_add :\n    primesBetween x (x+y) \u2264 (primeInterSieve x y z hz).siftedSum + z := by\n  classical\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a (Finset.Icc 1 (Nat.floor z))).card\n  \u00b7 rw[primesBetween]\n    norm_cast\n    apply Finset.card_le_card\n    apply primesBetween_subset _ _ _ hx\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card\n    + \u2191(Finset.Icc 1 (Nat.floor z)).card\n  \u00b7 norm_cast\n    apply Finset.card_union_le\n  rw[siftedSum_eq_card]\n  gcongr\n  rw[Nat.card_Icc]\n  simp\n  apply Nat.floor_le\n  linarith\n\nsection Remainder\n\n", "theoremStatement": "theorem Ioc_filter_dvd_eq (d a b: \u2115) (hd : d \u2260 0) :\n  Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b) =\n    Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d)) ", "theoremName": "BrunTitchmarsh.Ioc_filter_dvd_eq", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/BrunTitchmarsh.lean", "module": 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"Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  ext n\n  simp only [Finset.mem_filter, Finset.mem_Ioc, Nat.ceil_le, Finset.mem_image,\n    Nat.le_floor_iff (show 0 \u2264 x+y by linarith)]\n  constructor\n  \u00b7 intro hn\n    use  n/d\n    rcases hn with \u27e8\u27e8han, hnb\u27e9, hd\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 exact Nat.div_lt_div_of_lt_of_dvd hd han\n    \u00b7 exact Nat.div_le_div_right (Nat.le_floor hnb)\n    \u00b7 exact Nat.div_mul_cancel hd\n  \u00b7 rintro \u27e8r, \u27e8ha, ha'\u27e9, rfl\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 refine (Nat.div_lt_iff_lt_mul ?_).mp ha\n      omega\n    \u00b7 exact Nat.mul_le_of_le_div d r b ha'\n    \u00b7 exact Nat.dvd_mul_left d r", "proofType": "tactic", "proofLengthLines": 17, "proofLengthTokens": 567}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.NumberTheory.PrimeCounting\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Data.Complex.ExponentialBounds\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\n/-!\n# Bounds for the Selberg sieve\nThis file proves a number of results to help bound `Sieve.selbergSum`\n\n## Main Results\n* `selbergBoundingSum_ge_sum_div`: If `\u03bd` is completely multiplicative then `S \u2265 \u2211_{n \u2264 \u221ay}, \u03bd n`\n* `boundingSum_ge_log`: If `\u03bd n = 1 / n` then `S \u2265 log y / 2`\n* `rem_sum_le_of_const`: If `R_d \u2264 C` then the error term is at most `C * y * (1 + log y)^3`\n-/\n\nopen scoped Nat ArithmeticFunction BigOperators Classical\n\nnoncomputable section\nnamespace Sieve\n\nlemma prodDistinctPrimes_squarefree (s : Finset \u2115) (h : \u2200 p \u2208 s, p.Prime) :\n    Squarefree (\u220f p in s, p) := by\n  refine Iff.mpr Nat.squarefree_iff_prime_squarefree ?_\n  intro p hp; by_contra h_dvd\n  by_cases hps : p \u2208 s\n  \u00b7 rw [\u2190Finset.mul_prod_erase (a:=p) (h := hps), mul_dvd_mul_iff_left (Nat.Prime.ne_zero hp)] at h_dvd\n    cases' Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) h_dvd with q hq\n    rw [Finset.mem_erase] at hq\n    exact hq.1.1 $ symm $ (Nat.prime_dvd_prime_iff_eq hp (h q hq.1.2)).mp hq.2\n  \u00b7 have : p \u2223 \u220f p in s, p := Trans.trans (dvd_mul_right p p) h_dvd\n    cases' Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) this with q hq\n    have heq : p = q := by\n      rw [\u2190Nat.prime_dvd_prime_iff_eq hp (h q hq.1)]\n      exact hq.2\n    rw [heq] at hps; exact hps hq.1\n\nlemma primorial_squarefree (n : \u2115) : Squarefree (primorial n) := by\n  apply prodDistinctPrimes_squarefree\n  simp_rw [Finset.mem_filter];\n  exact fun _ h => h.2\n\ntheorem zeta_pos_of_prime : \u2200 (p : \u2115), Nat.Prime p \u2192 (0:\u211d) < (\u2191\u03b6:ArithmeticFunction \u211d) p := by\n  intro p hp\n  rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, if_neg (Nat.Prime.ne_zero hp)]\n  norm_num\n\ntheorem zeta_lt_self_of_prime : \u2200 (p : \u2115), Nat.Prime p \u2192 (\u2191\u03b6:ArithmeticFunction \u211d) p < (p:\u211d) := by\n  intro p hp\n  rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, if_neg (Nat.Prime.ne_zero hp)]\n  norm_num;\n  exact Nat.succ_le.mp (Nat.Prime.two_le hp)\n\ntheorem prime_dvd_primorial_iff (n p : \u2115) (hp : p.Prime) :\n    p \u2223 primorial n \u2194 p \u2264 n := by\n  unfold primorial\n  constructor\n  \u00b7 intro h\n    let h' : \u2203 i, i \u2208 Finset.filter Nat.Prime (Finset.range (n + 1)) \u2227 p \u2223 i := Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) h\n    cases' h' with q hq\n    rw [Finset.mem_filter, Finset.mem_range] at hq\n    rw [prime_dvd_prime_iff_eq (Nat.Prime.prime hp) (Nat.Prime.prime hq.1.2)] at hq\n    rw [hq.2]\n    exact Nat.lt_succ.mp hq.1.1\n  \u00b7 intro h\n    apply Finset.dvd_prod_of_mem\n    rw [Finset.mem_filter, Finset.mem_range]\n    exact \u27e8Nat.lt_succ.mpr h, hp\u27e9\n\ntheorem siftedSum_eq (s : SelbergSieve) (hw : \u2200 i \u2208 s.support, s.weights i = 1) (z : \u211d) (hz : 1 \u2264 z) (hP : s.prodPrimes = primorial (Nat.floor z)) :\n    s.siftedSum = (s.support.filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  dsimp only [Sieve.siftedSum]\n  rw [Finset.card_eq_sum_ones, \u2190Finset.sum_filter, Nat.cast_sum]\n  apply Finset.sum_congr;\n  rw [hP]\n  ext d; constructor\n  \u00b7 intro hd\n    rw [Finset.mem_filter] at *\n    constructor\n    \u00b7 exact hd.1\n    \u00b7 intro p hpp hpy\n      rw [\u2190Nat.Prime.coprime_iff_not_dvd hpp]\n      apply Nat.Coprime.coprime_dvd_left _ hd.2\n      rw [prime_dvd_primorial_iff _ _ hpp]\n      apply Nat.le_floor hpy\n  \u00b7 intro h\n    rw [Finset.mem_filter] at *\n    constructor\n    \u00b7 exact h.1\n    refine Nat.coprime_of_dvd ?_\n    intro p hp\n    erw [prime_dvd_primorial_iff _ _ hp]\n    intro hpy\n    apply h.2 p hp\n    trans \u2191(Nat.floor z)\n    \u00b7 norm_cast\n    \u00b7 apply Nat.floor_le\n      linarith only [hz]\n  simp_rw [Nat.cast_one]\n  intro x hx\n  simp only [Finset.filter_congr_decidable, Finset.mem_filter] at hx\n  apply hw x hx.1\n\ndef CompletelyMultiplicative (f : ArithmeticFunction \u211d) : Prop := f 1 = 1 \u2227 \u2200 a b, f (a*b) = f a * f b\n\nnamespace CompletelyMultiplicative\nopen ArithmeticFunction\ntheorem zeta : CompletelyMultiplicative \u03b6 := by\n  unfold CompletelyMultiplicative\n  simp_rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, ite_false, Nat.cast_one,\n    mul_eq_zero, Nat.cast_ite, CharP.cast_eq_zero, mul_ite, mul_zero, true_and]\n  intro a b;\n  by_cases ha : a = 0\n  \u00b7 simp [ha]\n  by_cases hb : b = 0\n  \u00b7 simp [hb]\n  rw [if_neg, if_neg hb, if_neg ha]; ring\n  push_neg; exact \u27e8ha, hb\u27e9\n\ntheorem id : CompletelyMultiplicative ArithmeticFunction.id := by\n    constructor <;> simp\n\ntheorem pmul (f g : ArithmeticFunction \u211d) (hf : CompletelyMultiplicative f) (hg : CompletelyMultiplicative g) :\n    CompletelyMultiplicative (ArithmeticFunction.pmul f g) := by\n  constructor\n  \u00b7 rw [pmul_apply, hf.1, hg.1, mul_one]\n  intro a b\n  simp_rw [pmul_apply, hf.2, hg.2]; ring\n\ntheorem pdiv {f g : ArithmeticFunction \u211d} (hf : CompletelyMultiplicative f) (hg : CompletelyMultiplicative g) :\n    CompletelyMultiplicative (ArithmeticFunction.pdiv f g) := by\n  constructor\n  \u00b7 rw [pdiv_apply, hf.1, hg.1, div_one]\n  intro a b\n  simp_rw [pdiv_apply, hf.2, hg.2]; ring\n\ntheorem isMultiplicative {f : ArithmeticFunction \u211d} (hf : CompletelyMultiplicative f) :\n    ArithmeticFunction.IsMultiplicative f :=\n  \u27e8hf.1, fun _ => hf.2 _ _\u27e9\n\ntheorem apply_pow (f : ArithmeticFunction \u211d) (hf : CompletelyMultiplicative f) (a n : \u2115) :\n    f (a^n) = f a ^ n := by\n  induction n with\n  | zero => simp_rw [Nat.zero_eq, pow_zero, hf.1]\n  | succ n' ih => simp_rw [pow_succ, hf.2, ih]\n\nend CompletelyMultiplicative\n\ntheorem prod_factors_one_div_compMult_ge (M : \u2115) (f : ArithmeticFunction \u211d) (hf : CompletelyMultiplicative f)\n  (hf_nonneg : \u2200 n, 0 \u2264 f n) (d : \u2115)  (hd : Squarefree d)  (hf_size : \u2200n, n.Prime \u2192 n \u2223 d \u2192 f n < 1):\n    f d * \u220f p in d.primeFactors, 1 / (1 - f p)\n    \u2265 \u220f p in d.primeFactors, \u2211 n in Finset.Icc 1 M, f (p^n) := by\n  calc f d * \u220f p in d.primeFactors, 1 / (1 - f p)\n    = \u220f p in d.primeFactors, f p / (1 - f p)                 := by\n        conv => { lhs; congr; rw [\u2190Nat.prod_primeFactors_of_squarefree hd] }\n        rw [hf.isMultiplicative.map_prod_of_subset_primeFactors _ _ subset_rfl,\n          \u2190Finset.prod_mul_distrib]\n        simp_rw[one_div, div_eq_mul_inv]\n  _ \u2265 \u220f p in d.primeFactors, \u2211 n in Finset.Icc 1 M, (f p)^n  := by\n    gcongr with p hp\n    \u00b7 exact fun p _ => Finset.sum_nonneg fun n _ => pow_nonneg (hf_nonneg p) n\n    rw [Nat.mem_primeFactors_of_ne_zero hd.ne_zero] at hp\n    rw [\u2190Nat.Ico_succ_right, geom_sum_Ico, \u2190mul_div_mul_left (c:= (-1:\u211d)) (f p ^ Nat.succ M - f p ^ 1)]\n    gcongr\n    \u00b7 apply hf_nonneg\n    \u00b7 linarith [hf_size p hp.1 hp.2]\n    \u00b7 rw [pow_one]\n      have : 0 \u2264 f p ^ (M.succ) := by\n        apply pow_nonneg\n        apply hf_nonneg\n      linarith only [this]\n    \u00b7 linarith only\n    \u00b7 norm_num\n    \u00b7 apply ne_of_lt $ hf_size p hp.1 hp.2\n    \u00b7 apply Nat.succ_le_iff.mpr (Nat.succ_pos _)\n\n  _ = \u220f p in d.primeFactors, \u2211 n in Finset.Icc 1 M, f (p^n)  := by\n     simp_rw [hf.apply_pow]\n\ntheorem prod_factors_sum_pow_compMult (M : \u2115) (hM : M \u2260 0) (f : ArithmeticFunction \u211d) (hf : CompletelyMultiplicative f) (d : \u2115) (hd : Squarefree d):\n    \u220f p in d.primeFactors, \u2211 n in Finset.Icc 1 M, f (p^n)\n    = \u2211 m in (d^M).divisors.filter (d \u2223 \u00b7), f m := by\n  rw [Finset.prod_sum]\n  let i : (a:_) \u2192 (ha : a \u2208 Finset.pi d.primeFactors fun p => Finset.Icc 1 M) \u2192 \u2115 :=\n    fun a _ => \u220f p in d.primeFactors.attach, p.1 ^ (a p p.2)\n  have hfact_i : \u2200 a ha,\n      \u2200 p , Nat.factorization (i a ha) p = if hp : p \u2208 d.primeFactors then a p hp else 0 := by\n    intro a ha p\n    by_cases hp : p \u2208 d.primeFactors\n    \u00b7 rw [dif_pos hp, Nat.factorization_prod, Finset.sum_apply',\n        Finset.sum_eq_single \u27e8p, hp\u27e9, Nat.factorization_pow, Finsupp.smul_apply,\n        Nat.Prime.factorization_self (Nat.prime_of_mem_factors $ List.mem_toFinset.mp hp)]\n      \u00b7 ring\n      \u00b7 intro q _ hq\n        rw [Nat.factorization_pow, Finsupp.smul_apply, smul_eq_zero]; right\n        apply Nat.factorization_eq_zero_of_not_dvd\n        rw [Nat.Prime.dvd_iff_eq, \u2190 exists_eq_subtype_mk_iff]; push_neg\n        exact fun _ => hq\n        exact Nat.prime_of_mem_factors $ List.mem_toFinset.mp q.2\n        exact (Nat.prime_of_mem_factors $ List.mem_toFinset.mp hp).ne_one\n      \u00b7 intro h\n        exfalso\n        exact h (Finset.mem_attach _ _)\n      \u00b7 exact fun q _ => pow_ne_zero _ (ne_of_gt (Nat.pos_of_mem_factors (List.mem_toFinset.mp q.2)))\n    \u00b7 rw [dif_neg hp]\n      by_cases hpp : p.Prime\n      swap\n      \u00b7 apply Nat.factorization_eq_zero_of_non_prime _ hpp\n      apply Nat.factorization_eq_zero_of_not_dvd\n      intro hp_dvd\n      obtain \u27e8\u27e8q, hq\u27e9, _, hp_dvd_pow\u27e9 := Prime.exists_mem_finset_dvd hpp.prime hp_dvd\n      apply hp\n      rw [Nat.mem_primeFactors]\n      constructor\n      \u00b7 exact hpp\n      refine \u27e8?_, hd.ne_zero\u27e9\n      trans q\n      \u00b7 apply Nat.Prime.dvd_of_dvd_pow hpp hp_dvd_pow\n      \u00b7 apply Nat.dvd_of_mem_factors $ List.mem_toFinset.mp hq\n\n  have hi_ne_zero : \u2200 (a : _) (ha : a \u2208 Finset.pi d.primeFactors fun _p => Finset.Icc 1 M),\n      i a ha \u2260 0 := by\n    intro a ha\n    erw [Finset.prod_ne_zero_iff]\n    exact fun p _ => pow_ne_zero _ (ne_of_gt (Nat.pos_of_mem_factors (List.mem_toFinset.mp p.property)))\n  save\n  have hi : \u2200 (a : _) (ha : a \u2208 Finset.pi d.primeFactors fun _p => Finset.Icc 1 M),\n      i a ha \u2208 (d^M).divisors.filter (d \u2223 \u00b7) := by\n    intro a ha\n    rw [Finset.mem_filter, Nat.mem_divisors, \u2190Nat.factorization_le_iff_dvd hd.ne_zero (hi_ne_zero a ha),\n      \u2190Nat.factorization_le_iff_dvd (hi_ne_zero a ha) (pow_ne_zero _ hd.ne_zero)]\n    constructor; constructor\n    \u00b7 rw [Finsupp.le_iff]; intro p _;\n      rw [hfact_i a ha]\n      by_cases hp :  p \u2208 d.primeFactors\n      \u00b7 rw [dif_pos hp]\n        rw [Nat.factorization_pow, Finsupp.smul_apply]\n        simp_rw [Finset.mem_pi, Finset.mem_Icc] at ha\n        trans (M \u2022 1)\n        \u00b7 norm_num;\n          exact (ha p hp).2\n        \u00b7 gcongr\n          rw [Nat.mem_primeFactors_of_ne_zero hd.ne_zero] at hp\n          rw [\u2190Nat.Prime.dvd_iff_one_le_factorization hp.1 hd.ne_zero]\n          exact hp.2\n      \u00b7 rw [dif_neg hp]; norm_num\n    \u00b7 apply pow_ne_zero _ hd.ne_zero\n    \u00b7 rw [Finsupp.le_iff]; intro p hp\n      rw [Nat.support_factorization] at hp\n      rw [hfact_i a ha]\n      rw [dif_pos hp]\n      trans 1\n      \u00b7 exact hd.natFactorization_le_one p\n      simp_rw [Finset.mem_pi, Finset.mem_Icc] at ha\n      exact (ha p hp).1\n\n  save\n  have h : \u2200 (a : _) (ha : a \u2208 Finset.pi d.primeFactors fun _p => Finset.Icc 1 M),\n      \u220f p in d.primeFactors.attach, f (p.1 ^ (a p p.2)) = f (i a ha) := by\n    intro a ha\n    apply symm\n    apply hf.isMultiplicative.map_prod\n    intro x _ y _ hxy\n    simp_rw [Finset.mem_pi, Finset.mem_Icc, Nat.succ_le] at ha\n    apply (Nat.coprime_pow_left_iff (ha x x.2).1 ..).mpr\n    apply (Nat.coprime_pow_right_iff (ha y y.2).1 ..).mpr\n    have hxp := Nat.prime_of_mem_factors (List.mem_toFinset.mp x.2)\n    rw [Nat.Prime.coprime_iff_not_dvd hxp]\n    rw [Nat.prime_dvd_prime_iff_eq hxp $ Nat.prime_of_mem_factors (List.mem_toFinset.mp y.2)]\n    exact fun hc => hxy (Subtype.eq hc)\n\n  save\n  have i_inj : \u2200 a ha b hb, i a ha = i b hb \u2192 a = b := by\n    intro a ha b hb hiab\n    apply_fun Nat.factorization at hiab\n    ext p hp\n    obtain hiabp := DFunLike.ext_iff.mp hiab p\n    rw [hfact_i a ha, hfact_i b hb, dif_pos hp, dif_pos hp] at hiabp\n    exact hiabp\n\n  save\n  have i_surj : \u2200 (b : \u2115), b \u2208 (d^M).divisors.filter (d \u2223 \u00b7) \u2192 \u2203 a ha, i a ha = b := by\n    intro b hb\n    have h : (fun p _ => (Nat.factorization b) p) \u2208 Finset.pi d.primeFactors fun p => Finset.Icc 1 M := by\n      rw [Finset.mem_pi]; intro p hp\n      rw [Finset.mem_Icc]\n      -- erw [List.mem_toFinset] at hp\n      rw [Finset.mem_filter] at hb\n      have hb_ne_zero : b \u2260 0 := ne_of_gt $ Nat.pos_of_mem_divisors hb.1\n      have hpp : p.Prime := Nat.prime_of_mem_primeFactors hp\n      constructor\n      \u00b7 rw [\u2190Nat.Prime.dvd_iff_one_le_factorization hpp hb_ne_zero]\n        \u00b7 exact Trans.trans (Nat.dvd_of_mem_primeFactors hp) hb.2\n      \u00b7 rw [Nat.mem_divisors] at hb\n        trans Nat.factorization (d^M) p\n        \u00b7 exact (Nat.factorization_le_iff_dvd hb_ne_zero hb.left.right).mpr hb.left.left p\n        rw [Nat.factorization_pow, Finsupp.smul_apply, smul_eq_mul]\n        have : d.factorization p \u2264 1 := by\n          apply hd.natFactorization_le_one\n        exact (mul_le_iff_le_one_right (Nat.pos_of_ne_zero hM)).mpr this\n    use (fun p _ => Nat.factorization b p)\n    use h\n    apply Nat.eq_of_factorization_eq\n    \u00b7 apply hi_ne_zero _ h\n    \u00b7 exact ne_of_gt $ Nat.pos_of_mem_divisors (Finset.mem_filter.mp hb).1\n    intro p\n    rw [hfact_i (fun p _ => (Nat.factorization b) p) h p]\n    rw [Finset.mem_filter, Nat.mem_divisors] at hb\n    by_cases hp : p \u2208 d.primeFactors\n    \u00b7 rw [dif_pos hp]\n    \u00b7 rw [dif_neg hp, eq_comm, Nat.factorization_eq_zero_iff, \u2190or_assoc]\n      rw [Nat.mem_primeFactors] at hp\n      left\n      push_neg at hp\n      by_cases hpp : p.Prime\n      \u00b7 right; intro h\n        apply absurd (hp hpp)\n        push_neg\n        exact \u27e8hpp.dvd_of_dvd_pow (h.trans hb.1.1), hd.ne_zero\u27e9\n      \u00b7 left; exact hpp\n\n  exact Finset.sum_bij i hi i_inj i_surj h\n\ntheorem prod_primes_dvd_of_dvd (P : \u2115) {s : Finset \u2115} (h : \u2200 p \u2208 s, p \u2223 P) (h' : \u2200 p \u2208 s, p.Prime):\n    \u220f p in s, p \u2223 P := by\n  simp_rw [Nat.prime_iff] at h'\n  apply Finset.prod_primes_dvd _ h' h\n\n", "theoremStatement": "lemma sqrt_le_self (x : \u211d) (hx : 1 \u2264 x) : Real.sqrt x \u2264 x ", "theoremName": "Sieve.sqrt_le_self", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/SelbergBounds.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds.jsonl", "positionMetadata": {"lineInFile": 335, "tokenPositionInFile": 13468, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 137, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", 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"Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Prime", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Int.ModEq", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", 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"Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Tactic.ArithMult.Init", 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"PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  refine Iff.mpr Real.sqrt_le_iff ?_\n  constructor\n  \u00b7 linarith\n  refine le_self_pow hx ?right.h\n  norm_num", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 113}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\n", "theoremStatement": "lemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) ", "theoremName": "Complex.one_div_cpow_eq", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "c237a4983cc26b38ba4b7e8f7b6a23fd2b36832a", "date": "2024-04-04"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": "PrimeNumberTheoremAnd.ZetaBounds.jsonl", "positionMetadata": 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"Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 118}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\n", "theoremStatement": "lemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x ", "theoremName": "nabla_log_main", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "a5d62dca38dbc573595247d8289c94dc78c50876", "date": "2024-03-12"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 690, "tokenPositionInFile": 34883, "theoremPositionInFile": 67}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 190, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", 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"Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 253}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\n", "theoremStatement": "lemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u ", "theoremName": "BoundedAtFilter.add_const", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "d73d873f6665eb482555ddcc45f9fadb39ddb8fd", "date": "2024-03-12"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 553, "tokenPositionInFile": 26816, "theoremPositionInFile": 48}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 83, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", 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"Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 182}}
+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\ntheorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) := by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz\n\ntheorem primesBetween_le_siftedSum_add :\n    primesBetween x (x+y) \u2264 (primeInterSieve x y z hz).siftedSum + z := by\n  classical\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a (Finset.Icc 1 (Nat.floor z))).card\n  \u00b7 rw[primesBetween]\n    norm_cast\n    apply Finset.card_le_card\n    apply primesBetween_subset _ _ _ hx\n  trans \u2191((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card\n    + \u2191(Finset.Icc 1 (Nat.floor z)).card\n  \u00b7 norm_cast\n    apply Finset.card_union_le\n  rw[siftedSum_eq_card]\n  gcongr\n  rw[Nat.card_Icc]\n  simp\n  apply Nat.floor_le\n  linarith\n\nsection Remainder\n\ntheorem Ioc_filter_dvd_eq (d a b: \u2115) (hd : d \u2260 0) :\n  Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b) =\n    Finset.image (fun x => x * d) (Finset.Ioc (a / d) (b / d)) := by\n  ext n\n  simp only [Finset.mem_filter, Finset.mem_Ioc, Nat.ceil_le, Finset.mem_image,\n    Nat.le_floor_iff (show 0 \u2264 x+y by linarith)]\n  constructor\n  \u00b7 intro hn\n    use  n/d\n    rcases hn with \u27e8\u27e8han, hnb\u27e9, hd\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 exact Nat.div_lt_div_of_lt_of_dvd hd han\n    \u00b7 exact Nat.div_le_div_right (Nat.le_floor hnb)\n    \u00b7 exact Nat.div_mul_cancel hd\n  \u00b7 rintro \u27e8r, \u27e8ha, ha'\u27e9, rfl\u27e9\n    refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n    \u00b7 refine (Nat.div_lt_iff_lt_mul ?_).mp ha\n      omega\n    \u00b7 exact Nat.mul_le_of_le_div d r b ha'\n    \u00b7 exact Nat.dvd_mul_left d r\n\ntheorem card_Ioc_filter_dvd (d a b: \u2115) (hd : d \u2260 0) :\n    (Finset.filter (fun x => d \u2223 x) (Finset.Ioc a b)).card = b / d - a / d  := by\n  rw [Ioc_filter_dvd_eq _ _ _ hd]\n  rw [Finset.card_image_of_injective _ <| mul_left_injective\u2080 hd]\n  simp\n\ntheorem multSum_eq (d : \u2115) (hd : d \u2260 0):\n    (primeInterSieve x y z hz).multSum d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) := by\n  unfold Sieve.multSum\n  rw[primeInterSieve]\n  simp\n  trans \u2191(Finset.Ioc (Nat.ceil x - 1) (Nat.floor (x+y)) |>.filter (d \u2223 \u00b7) |>.card)\n  \u00b7 rw [\u2190Nat.Icc_succ_left]\n    congr\n    rw [\u2190Nat.succ_sub]; rfl\n    simp [hx]\n  \u00b7 rw[BrunTitchmarsh.card_Ioc_filter_dvd _ _ _ hd]\n\ntheorem rem_eq (d : \u2115) (hd : d \u2260 0) : (primeInterSieve x y z hz).rem d = \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) - (\u2191d)\u207b\u00b9 * y := by\n  unfold Sieve.rem\n  rw[multSum_eq x y z hx hz d hd]\n  simp [primeInterSieve, if_neg hd]\n\ntheorem Nat.ceil_le_self_add_one (x : \u211d) (hx : 0 \u2264 x) : Nat.ceil x \u2264 x + 1 := by\n  trans Nat.floor x + 1\n  \u00b7 norm_cast\n    exact Nat.ceil_le_floor_add_one x\n  gcongr\n  apply Nat.floor_le hx\n\ntheorem floor_approx (x : \u211d) (hx : 0 \u2264 x) : \u2203 C, |C| \u2264 1 \u2227  \u2191((Nat.floor x)) = x + C := by\n  use \u2191(Nat.floor x) - x\n  simp\n  rw[abs_le]\n  constructor\n  \u00b7 simp only [neg_le_sub_iff_le_add]\n    linarith [Nat.lt_floor_add_one x]\n  \u00b7 simp only [tsub_le_iff_right]\n    linarith [Nat.floor_le hx]\n\ntheorem ceil_approx (x : \u211d) (hx : 0 \u2264 x) : \u2203 C, |C| \u2264 1 \u2227  \u2191((Nat.ceil x)) = x + C := by\n  use \u2191(Nat.ceil x) - x\n  simp\n  rw[abs_le]\n  constructor\n  \u00b7 simp only [neg_le_sub_iff_le_add]\n    linarith [Nat.le_ceil x]\n  \u00b7 simp only [tsub_le_iff_right]\n    rw[add_comm]\n    exact Nat.ceil_le_self_add_one x hx\n\ntheorem nat_div_approx (a b : \u2115) : \u2203 C, |C| \u2264 1 \u2227 \u2191(a/b) = (a/b : \u211d) + C := by\n  rw[\u2190Nat.floor_div_eq_div (\u03b1:=\u211d)]\n  apply floor_approx (a/b:\u211d) (by positivity)\n\ntheorem floor_div_approx (x : \u211d) (hx : 0 \u2264 x) (d : \u2115) : \u2203 C, |C| \u2264 2 \u2227  \u2191((Nat.floor x)/d) = x / d + C := by\n  by_cases hd : d = 0\n  \u00b7 simp [hd]\n  obtain \u27e8C\u2081, hC\u2081_le, hC\u2081\u27e9 := nat_div_approx (Nat.floor x) d\n  obtain \u27e8C\u2082, hC\u2082_le, hC\u2082\u27e9 := floor_approx x hx\n  rw[hC\u2081, hC\u2082]\n  use  C\u2081 + C\u2082/d\n  refine \u27e8?_, by ring\u27e9\n  have : |C\u2081 + C\u2082/d| \u2264 |C\u2081| + |C\u2082/d| := abs_add C\u2081 (C\u2082 / \u2191d)\n  have : |C\u2082/d| \u2264 |C\u2082| := by\n    rw[abs_div]\n    apply div_le_self\n    \u00b7 exact abs_nonneg C\u2082\n    \u00b7 simp only [Nat.abs_cast, Nat.one_le_cast]\n      omega\n  linarith\n\ntheorem abs_rem_le {d : \u2115} (hd : d \u2260 0) : |(primeInterSieve x y z hz).rem d| \u2264 5 := by\n  rw[rem_eq _ _ _ hx hz _ hd]\n\n  have hpush : \u2191(\u230ax + y\u230b\u208a / d - (\u2308x\u2309\u208a - 1) / d) = ( \u2191(\u230ax + y\u230b\u208a / d) - \u2191((\u2308x\u2309\u208a - 1) / d) : \u211d) := by\n    rw [Nat.cast_sub]\n    gcongr\n    rw[Nat.le_floor_iff]\n    rw[\u2190add_le_add_iff_right 1]\n    norm_cast\n    rw [Nat.sub_add_cancel (by simp [hx])]\n    linarith [Nat.ceil_le_self_add_one x (le_of_lt hx)]\n    linarith\n\n  rw[hpush]\n  obtain \u27e8C\u2081, hC\u2081_le, hC\u2081\u27e9 := floor_div_approx (x + y) (by linarith) d\n  obtain \u27e8C\u2082, hC\u2082_le, hC\u2082\u27e9 := nat_div_approx (Nat.ceil x - 1) d\n  obtain \u27e8C\u2083, hC\u2083_le, hC\u2083\u27e9 := ceil_approx (x) (by linarith)\n  rw[hC\u2081, hC\u2082, Nat.cast_sub, hC\u2083]\n  ring_nf\n  have : |(\u2191d)\u207b\u00b9 - (\u2191d)\u207b\u00b9 * C\u2083 + (C\u2081 - C\u2082)| \u2264 |(\u2191d)\u207b\u00b9 - (\u2191d)\u207b\u00b9*C\u2083| + |C\u2081 - C\u2082| := by\n    apply (abs_add _ _)\n  have : |(\u2191d)\u207b\u00b9 - (\u2191d)\u207b\u00b9*C\u2083| \u2264 |(\u2191d)\u207b\u00b9| + |(\u2191d)\u207b\u00b9*C\u2083| := abs_sub _ _\n  have : |C\u2081 - C\u2082| \u2264 |C\u2081| + |C\u2082| := abs_sub _ _\n  have : |(d:\u211d)\u207b\u00b9| \u2264 1 := by\n    rw[abs_inv]\n    simp only [Nat.abs_cast]\n    apply Nat.cast_inv_le_one\n  have : |(\u2191d)\u207b\u00b9*C\u2083| \u2264 |C\u2083| := by\n    rw[inv_mul_eq_div, abs_div]\n    apply div_le_self\n    \u00b7 exact abs_nonneg _\n    \u00b7 simp only [Nat.abs_cast, Nat.one_le_cast]\n      omega\n  linarith\n  \u00b7 simp [hx]\n\nend Remainder\n\ntheorem boudingSum_ge : (primeInterSieve x y z hz).selbergBoundingSum \u2265 Real.log z / 2 := by\n  apply boundingSum_ge_log\n  \u00b7 rfl\n  \u00b7 intro p hpp hp\n    erw [prime_dvd_primorial_iff]\n    apply Nat.le_floor\n    exact hp\n    exact hpp\n\ntheorem primeSieve_rem_sum_le :\n    \u2211 d in (primeInterSieve x y z hz).prodPrimes.divisors, (if (d : \u211d) \u2264 z then (3:\u211d) ^ \u03c9 d * |(primeInterSieve x y z hz).rem d| else 0)\n      \u2264 5 * z * (1+Real.log z)^3 := by\n  apply rem_sum_le_of_const (primeInterSieve x y z hz) 5 ?_\n  intro d hd\n  apply abs_rem_le <;> linarith\n\ntheorem siftedSum_le (hz : 1 < z) :\n    (primeInterSieve x y z (le_of_lt hz)).siftedSum \u2264 2 * y / Real.log z + 5 * z * (1+Real.log z)^3  := by\n  apply le_trans (SelbergSieve.selberg_bound_simple ..)\n  calc _ \u2264 y / (Real.log z / 2) + 5 * z * (1+Real.log z)^3 := ?_\n       _ = _ := by ring\n  gcongr\n  \u00b7 linarith [Real.log_pos hz]\n  \u00b7 rfl\n  \u00b7 apply boudingSum_ge\n  \u00b7 apply primeSieve_rem_sum_le x y z hx hy\n\ntheorem primesBetween_le (hz : 1 < z) :\n    primesBetween x (x+y) \u2264 2 * y / Real.log z + 6 * z * (1+Real.log z)^3 := by\n  have : z \u2264 z * (1+Real.log z)^3 := by\n    apply le_mul_of_one_le_right\n    \u00b7 linarith\n    apply one_le_pow_of_one_le _ _\n    linarith [Real.log_nonneg (by linarith)]\n  linarith [siftedSum_le _ _ _ hx hy hz, primesBetween_le_siftedSum_add x y z hx (le_of_lt hz)]\n\ntheorem primesBetween_one (n : \u2115) : primesBetween 1 n = ((Finset.range (n+1)).filter Nat.Prime).card := by\n  rw [primesBetween]\n  congr 1\n  ext p\n  simp only [Nat.ceil_one, Nat.floor_coe, Finset.mem_filter, Finset.mem_Icc, Finset.mem_range,\n    and_congr_left_iff]\n  intro hp\n  refine \u27e8?_, ?_\u27e9\n  \u00b7 exact fun h => by omega\n  \u00b7 refine fun h => \u27e8by have := hp.pos; omega, by omega\u27e9\n\ntheorem primesBetween_mono_right (a b c : \u211d) (hbc : b \u2264 c) : primesBetween a b \u2264 primesBetween a c := by\n  dsimp only [primesBetween]\n  apply Finset.card_le_card\n  intro p\n  simp only [Finset.mem_filter, Finset.mem_Icc, Nat.ceil_le, and_imp]\n  intro ha hb hp\n  refine \u27e8\u27e8ha, hb.trans (Nat.floor_mono hbc)\u27e9, hp\u27e9\n\ntheorem tmp (N : \u2115) : ((Finset.range N).filter Nat.Prime).card \u2264 4 * (N / Real.log N) + 6 *(N ^ (1/2 : \u211d) * (1 + 1/2 * Real.log N)^3) := by\n  trans \u2191((Finset.range (N+1)).filter Nat.Prime).card\n  \u00b7 norm_cast\n    apply Finset.card_le_card\n    intro n\n    simp\n    refine fun hnN hp \u21a6 \u27e8by omega, hp\u27e9\n  rw [\u2190 primesBetween_one]\n  by_cases hN : N = 0\n  \u00b7 simp [hN, primesBetween]\n  by_cases hN : N = 1\n  \u00b7 simp (config := {decide:=true}) [hN, primesBetween]\n  have h : primesBetween 1 (1 + N) \u2264\n      2 * (N / Real.log (N^(1/2:\u211d))) + 6 * (N ^ (1 / 2 : \u211d) * (1 + Real.log (N ^ (1 / 2 : \u211d))) ^ 3) := by\n    convert (primesBetween_le 1 N (N ^ (1/2 : \u211d)) (by norm_num) (by norm_cast; omega)\n      (Real.one_lt_rpow (by norm_cast; omega) (by norm_num))) using 1\n    ring\n  calc\n    _ \u2264 (primesBetween 1 (1+N):\u211d) := by\n      norm_cast; apply primesBetween_mono_right; norm_cast; omega\n    _ \u2264 _ := by\n      rw [Real.log_rpow (by norm_num; omega)] at h\n      convert h using 2\n      ring\n\ntheorem rpow_mul_rpow_log_isBigO_id_div_log (k : \u211d) {r : \u211d} (hr : r < 1) : (fun x \u21a6 (x : \u211d) ^ (r : \u211d) * (Real.log x)^k) =O[atTop] (fun x \u21a6 x / Real.log x) := calc\n  (fun x \u21a6 (x : \u211d) ^ (r : \u211d) * (Real.log x)^k) =O[atTop] (fun x \u21a6 (x : \u211d) ^ (r : \u211d) * x ^ ((1-r)/2 : \u211d)) := by\n    apply IsBigO.mul (isBigO_refl ..)\n    apply (isLittleO_log_rpow_rpow_atTop k (by linarith) ..).isBigO\n  _ =\u1da0[atTop] (fun N \u21a6 (N : \u211d) * (N ^ ((1-r)/2 : \u211d))\u207b\u00b9) := by\n    filter_upwards [Filter.eventually_gt_atTop 0]\n    intro N hN\n    trans (N ^ (1 : \u211d) * (N ^ ((1-r)/2 : \u211d))\u207b\u00b9)\n    \u00b7 rw [\u2190 Real.rpow_add hN, \u2190 Real.rpow_neg hN.le, \u2190 Real.rpow_add hN]\n      ring_nf\n    \u00b7 rw [\u2190 Nat.cast_one, Real.rpow_nat_cast, pow_one]\n  _ =O[atTop] (fun N \u21a6 (N : \u211d) * (Real.log N)\u207b\u00b9) := by\n    apply IsBigO.mul (isBigO_refl ..)\n    apply IsBigO.inv_rev\n    apply (isLittleO_log_rpow_atTop (by linarith) ..).isBigO\n    \u00b7 filter_upwards [Filter.eventually_gt_atTop 1]\n      intro N hN hcontra\n      linarith [Real.log_pos hN]\n  _ = (fun N \u21a6 (N : \u211d)/(Real.log N)) := by\n    simp_rw [div_eq_mul_inv]\n\n", "theoremStatement": "theorem err_isBigO : (fun x \u21a6 (x ^ (1 / 2 : \u211d) * (1 + 1 / 2 * Real.log x) ^ 3)) =O[atTop] fun x \u21a6 (x / Real.log x) ", "theoremName": "BrunTitchmarsh.err_isBigO", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "f104ed74a1acd23285ae4b9b1734f6316ea0748a", "date": "2024-04-03"}, 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"Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  calc\n    _ =O[atTop] (fun x \u21a6 x ^ (1/2:\u211d) * (Real.log x) ^ 3) := by\n      apply IsBigO.mul (isBigO_refl ..)\n      apply Real.isLittleO_const_log_atTop.isBigO.add ((isBigO_refl ..).const_mul_left ..) |>.pow\n    _ =O[atTop] _ := by\n      convert rpow_mul_rpow_log_isBigO_id_div_log 3 (?_) <;> norm_num\n      rw [\u2190 Real.rpow_nat_cast]\n      norm_cast", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 355}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module selberg\n-/\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic\n\n/-!\n# The Selberg Sieve\n\nThis file proves `selberg_bound_simple`, the main theorem of the Selberg.\n-/\n\nnoncomputable section\n\nopen scoped BigOperators Classical Sieve\n\nopen Finset Real Nat Sieve.UpperBoundSieve ArithmeticFunction Sieve\n\nstructure SelbergSieve extends Sieve where mk ::\n  level : \u211d\n  one_le_level : 1 \u2264 level\n\nnamespace SelbergSieve\nset_option quotPrecheck false\n\nvariable (s : SelbergSieve)\nlocal notation3 \"\u03bd\" => Sieve.nu (toSieve s)\nlocal notation3 \"P\" => Sieve.prodPrimes (toSieve s)\nlocal notation3 \"a\" => Sieve.weights (toSieve s)\nlocal notation3 \"X\" => Sieve.totalMass (toSieve s)\nlocal notation3 \"R\" => Sieve.rem (toSieve s)  -- this one seems broken\nlocal notation3 \"g\" => Sieve.selbergTerms (toSieve s)\nlocal notation3 \"y\" => SelbergSieve.level s\nlocal notation3 \"hy\" => SelbergSieve.one_le_level s\n\n--set_option profiler true\n@[simp]\ndef selbergBoundingSum : \u211d :=\n  \u2211 l in divisors P, if l ^ 2 \u2264 y then g l else 0\n\nset_option quotPrecheck false\nlocal notation3 \"S\" => SelbergSieve.selbergBoundingSum s\n\ntheorem selbergBoundingSum_pos :\n    0 < S := by\n  dsimp only [selbergBoundingSum]\n  rw [\u2190 sum_filter]\n  apply sum_pos;\n  \u00b7 intro l hl\n    rw [mem_filter, mem_divisors] at hl\n    \u00b7 apply s.selbergTerms_pos _ (hl.1.1)\n  \u00b7 simp_rw [Finset.Nonempty, mem_filter]; use 1\n    constructor\n    \u00b7 apply one_mem_divisors.mpr s.prodPrimes_ne_zero\n    rw [cast_one, one_pow]\n    exact s.one_le_level\n\ntheorem selbergBoundingSum_ne_zero : S \u2260 0 := by\n  apply _root_.ne_of_gt\n  exact s.selbergBoundingSum_pos\n\ntheorem selbergBoundingSum_nonneg : 0 \u2264 S := _root_.le_of_lt s.selbergBoundingSum_pos\n\ndef selbergWeights : \u2115 \u2192 \u211d := fun d =>\n  if d \u2223 P then\n    (\u03bd d)\u207b\u00b9 * g d * \u03bc d * S\u207b\u00b9 *\n      \u2211 m in divisors P, if (d * m) ^ 2 \u2264 y \u2227 m.Coprime d then g m else 0\n  else 0\n\n-- This notation traditionally uses \u03bb, which is unavailable in lean\nset_option quotPrecheck false\nlocal notation3 \"\u03b3\" => SelbergSieve.selbergWeights s\n\ntheorem selbergWeights_eq_zero_of_not_dvd {d : \u2115} (hd : \u00ac d \u2223 P) :\n    \u03b3 d = 0 := by\n  rw [selbergWeights, if_neg hd]\n\ntheorem selbergWeights_eq_zero (d : \u2115) (hd : \u00acd ^ 2 \u2264 y) :\n    \u03b3 d = 0 := by\n  dsimp only [selbergWeights]\n  split_ifs with h\n  \u00b7 rw [mul_eq_zero_of_right _]\n    apply Finset.sum_eq_zero\n    refine fun m hm => if_neg ?_\n    intro hyp\n    have : (d^2:\u211d) \u2264 (d*m)^2 := by\n      norm_cast;\n      refine Nat.pow_le_pow_of_le_left ?h 2\n      exact Nat.le_mul_of_pos_right _ (Nat.pos_of_mem_divisors hm)\n    linarith [hyp.1]\n  \u00b7 rfl\n\n@[aesop safe]\ntheorem selbergWeights_mul_mu_nonneg (d : \u2115) (hdP : d \u2223 P) :\n    0 \u2264 \u03b3 d * \u03bc d :=\n  by\n  have := s.selbergBoundingSum_nonneg\n  dsimp only [selbergWeights]\n  rw [if_pos hdP]; rw [mul_assoc]\n  trans ((\u03bc d :\u211d)^2 * (\u03bd d)\u207b\u00b9 * g d * S\u207b\u00b9 * \u2211 m in divisors P,\n          if (d * m) ^ 2 \u2264 y \u2227 Coprime m d then g m else 0)\n  swap; apply le_of_eq; ring\n  apply mul_nonneg; apply div_nonneg; apply mul_nonneg; apply mul_nonneg\n  \u00b7 apply sq_nonneg\n  \u00b7 rw [inv_nonneg]\n    exact le_of_lt $ s.nu_pos_of_dvd_prodPrimes hdP\n  \u00b7 exact le_of_lt $ s.selbergTerms_pos d hdP\n  \u00b7 exact s.selbergBoundingSum_nonneg\n  apply sum_nonneg; intro m hm\n  split_ifs with h\n  \u00b7 exact le_of_lt $ s.selbergTerms_pos m (dvd_of_mem_divisors hm)\n  \u00b7 rfl\n\nlemma sum_mul_subst (k n: \u2115) {f : \u2115 \u2192 \u211d} (h : \u2200 l, l \u2223 n \u2192 \u00ac k \u2223 l \u2192 f l = 0) :\n      \u2211 l in n.divisors, f l\n    = \u2211 m in n.divisors, if k*m \u2223 n then f (k*m) else 0 := by\n  by_cases hn: n = 0\n  \u00b7 simp [hn]\n  by_cases hkn : k \u2223 n\n  swap\n  \u00b7 rw [sum_eq_zero, sum_eq_zero]\n    \u00b7 rintro m _\n      rw [if_neg]\n      rintro h\n      apply hkn\n      exact (Nat.dvd_mul_right k m).trans h\n    \u00b7 intro l hl; apply h l (dvd_of_mem_divisors hl)\n      apply fun hkl => hkn <| hkl.trans (dvd_of_mem_divisors hl)\n  trans (\u2211 l in n.divisors, \u2211 m in n.divisors, if l=k*m then f l else 0)\n  \u00b7 rw [sum_congr rfl]; intro l hl\n    by_cases hkl : k \u2223 l\n    swap\n    \u00b7 rw [h l (dvd_of_mem_divisors hl) hkl, sum_eq_zero];\n      intro m _; rw [ite_id]\n    rw [sum_eq_single (l/k)]\n    \u00b7 rw[if_pos]; rw [Nat.mul_div_cancel' hkl]\n    \u00b7 intro m _ hmlk\n      apply if_neg; revert hmlk; contrapose!; intro hlkm\n      rw [hlkm, mul_comm, Nat.mul_div_cancel];\n      apply Nat.pos_of_dvd_of_pos hkn (Nat.pos_of_ne_zero hn)\n    \u00b7 contrapose!; intro _\n      rw [mem_divisors]\n      exact \u27e8Trans.trans (Nat.div_dvd_of_dvd hkl) (dvd_of_mem_divisors hl), hn\u27e9\n  \u00b7 rw [sum_comm, sum_congr rfl]; intro m _\n    split_ifs with hdvd\n    \u00b7 rw [\u2190Aux.sum_intro]\n      simp only [mem_divisors, hdvd, ne_eq, hn, not_false_eq_true, and_self]\n    \u00b7 apply sum_eq_zero; intro l hl\n      apply if_neg;\n      rintro rfl\n      simp only [mem_divisors, ne_eq] at hl\n      exact hdvd hl.1\n\n--Important facts about the selberg weights\ntheorem selbergWeights_eq_dvds_sum (d : \u2115) :\n    \u03bd d * \u03b3 d =\n      S\u207b\u00b9 * \u03bc d *\n        \u2211 l in divisors P, if d \u2223 l \u2227 l ^ 2 \u2264 y then g l else 0 := by\n  by_cases h_dvd : d \u2223 P\n  swap\n  \u00b7 dsimp only [selbergWeights]; rw [if_neg h_dvd]\n    rw [sum_eq_zero]; ring\n    intro l hl; rw [mem_divisors] at hl\n    rw [if_neg]; push_neg; intro h\n    exfalso; exact h_dvd (dvd_trans h hl.left)\n  dsimp only [selbergWeights]\n  rw [if_pos h_dvd]\n  repeat rw [mul_sum]\n  -- change of variables l=m*d\n  apply symm\n  rw [sum_mul_subst d P]\n  apply sum_congr rfl\n  intro m hm\n  rw [mul_ite_zero, \u2190ite_and, mul_ite_zero, mul_ite_zero]\n  apply if_ctx_congr _ _ fun _ => rfl\n  \u00b7 rw [coprime_comm]\n    constructor\n    \u00b7 intro h\n      push_cast at h\n      exact \u27e8h.2.2, coprime_of_squarefree_mul $ Squarefree.squarefree_of_dvd h.1 s.prodPrimes_squarefree\u27e9\n    \u00b7 intro h\n      push_cast\n      exact \u27e8 Coprime.mul_dvd_of_dvd_of_dvd h.2 h_dvd (dvd_of_mem_divisors hm), Nat.dvd_mul_right d m, h.1\u27e9\n  \u00b7 intro h\n    trans ((\u03bd d)\u207b\u00b9 * (\u03bd d) * g d * \u03bc d / S * g m)\n    \u00b7 rw [inv_mul_cancel (s.nu_ne_zero h_dvd), s.selbergTerms_mult.map_mul_of_coprime\n        $ coprime_comm.mp h.2]\n      ring\n    ring\n  \u00b7 intro l _ hdl\n    rw [if_neg, mul_zero]\n    push_neg; intro h; contradiction\n\ntheorem selbergWeights_diagonalisation (l : \u2115) (hl : l \u2208 divisors P) :\n    (\u2211 d in divisors P, if l \u2223 d then \u03bd d * \u03b3 d else 0) =\n      if l ^ 2 \u2264 y then g l * \u03bc l * S\u207b\u00b9 else 0 := by\n  calc\n    (\u2211 d in divisors P, if l \u2223 d then \u03bd d * \u03b3 d else 0) =\n        \u2211 d in divisors P, \u2211 k in divisors P,\n          if l \u2223 d \u2227 d \u2223 k \u2227 k ^ 2 \u2264 y then g k * S\u207b\u00b9 * (\u03bc d:\u211d) else 0 := by\n      apply sum_congr rfl; intro d _\n      rw [selbergWeights_eq_dvds_sum, \u2190 boole_mul, mul_sum, mul_sum]\n      apply sum_congr rfl; intro k _\n      rw [mul_ite_zero, ite_zero_mul_ite_zero]\n      apply if_ctx_congr Iff.rfl _ (fun _ => rfl);\n      intro _; ring\n    _ = \u2211 k in divisors P, if k ^ 2 \u2264 y then\n            (\u2211 d in divisors P, if l \u2223 d \u2227 d \u2223 k then (\u03bc d:\u211d) else 0) * g k * S\u207b\u00b9\n          else 0 := by\n      rw [sum_comm]; apply sum_congr rfl; intro k _\n      apply symm\n      rw [\u2190 boole_mul, sum_mul, sum_mul, mul_sum, sum_congr rfl]\n      intro d _\n      rw [ite_zero_mul, ite_zero_mul, ite_zero_mul, one_mul, \u2190ite_and]\n      apply if_ctx_congr _ _ (fun _ => rfl)\n      \u00b7 tauto\n      intro _; ring\n    _ = if l ^ 2 \u2264 y then g l * \u03bc l * S\u207b\u00b9 else 0 := by\n      rw [Aux.sum_intro (f:=fun _ => if l^2 \u2264 y then g l * \u03bc l * S\u207b\u00b9 else 0) (divisors P) l hl]\n      apply sum_congr rfl; intro k hk\n      rw [Aux.moebius_inv_dvd_lower_bound_real s.prodPrimes_squarefree l _ (dvd_of_mem_divisors hk),\n        \u2190ite_and, ite_zero_mul, ite_zero_mul, \u2190 ite_and]\n      apply if_ctx_congr _ _ fun _ => rfl\n      rw [and_comm, eq_comm]; apply and_congr_right\n      intro heq; rw [heq]\n      intro h; rw[h.1]; ring\n\ndef selbergMuPlus : \u2115 \u2192 \u211d :=\n  Sieve.lambdaSquared \u03b3\n\nset_option quotPrecheck false\nlocal notation3 \"\u03bc\u207a\" => SelbergSieve.selbergMuPlus s\n\ntheorem weight_one_of_selberg : \u03b3 1 = 1 := by\n  dsimp only [selbergWeights]\n  rw [if_pos (one_dvd P), s.nu_mult.left, s.selbergTerms_mult.left]\n  -- rw [ArithmeticFunction.moebius_apply_one, Int.cast_one]\n  simp only [inv_one, mul_one, isUnit_one, IsUnit.squarefree, moebius_apply_of_squarefree,\n    cardFactors_one, _root_.pow_zero, Int.cast_one, selbergBoundingSum, cast_pow, one_mul,\n    coprime_one_right_eq_true, and_true, cast_one]\n  rw [inv_mul_cancel]\n  convert s.selbergBoundingSum_ne_zero\n\ntheorem selberg\u03bcPlus_eq_zero (d : \u2115) (hd : \u00acd \u2264 y) : \u03bc\u207a d = 0 :=\n  by\n  apply Sieve.lambdaSquared_eq_zero_of_support _ y _ d hd\n  apply s.selbergWeights_eq_zero\n\ndef selbergUbSieve : UpperBoundSieve :=\n  \u27e8\u03bc\u207a, Sieve.upperMoebius_of_lambda_sq \u03b3 (s.weight_one_of_selberg)\u27e9\n\n-- proved for general lambda squared sieves\ntheorem mainSum_eq_diag_quad_form :\n    s.mainSum \u03bc\u207a =\n      \u2211 l in divisors P,\n        1 / g l *\n          (\u2211 d in divisors P, if l \u2223 d then \u03bd d * \u03b3 d else 0) ^ 2 :=\n  by apply lambdaSquared_mainSum_eq_diag_quad_form\n\n\n/-- These two are in Mathlib per #10672 -/\ntheorem moebius_sq_eq_one_of_squarefree {l : \u2115} (hl : Squarefree l) : \u03bc l ^ 2 = 1 := by\n  rw [moebius_apply_of_squarefree hl, \u2190 pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]\n\ntheorem abs_moebius_eq_one_of_squarefree {l : \u2115} (hl : Squarefree l) : |\u03bc l| = 1 := by\n  simp only [moebius_apply_of_squarefree hl, abs_pow, abs_neg, abs_one, one_pow]\n\n", "theoremStatement": "theorem selberg_bound_simple_mainSum :\n    s.mainSum \u03bc\u207a = S\u207b\u00b9 ", "theoremName": "SelbergSieve.selberg_bound_simple_mainSum", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/Selberg.lean", "module": 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"Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  by\n  rw [mainSum_eq_diag_quad_form]\n  trans (\u2211 l in divisors P, (if l ^ 2 \u2264 y then g l *  (S\u207b\u00b9) ^ 2 else 0))\n  \u00b7 apply sum_congr rfl; intro l hl\n    rw [s.selbergWeights_diagonalisation l hl, ite_pow, zero_pow, mul_ite_zero]\n    apply if_congr Iff.rfl _ rfl\n    trans (1/g l * g l * g l * (\u03bc l:\u211d)^2  * (S\u207b\u00b9) ^ 2)\n    \u00b7 ring\n    norm_cast; rw [moebius_sq_eq_one_of_squarefree $ s.squarefree_of_mem_divisors_prodPrimes hl]\n    rw [one_div_mul_cancel $ _root_.ne_of_gt $ s.selbergTerms_pos l $ dvd_of_mem_divisors hl]\n    ring\n    linarith\n  conv => {lhs; congr; {skip}; {ext i; rw [\u2190 ite_zero_mul]}}\n  dsimp only [selbergBoundingSum]\n  rw [\u2190sum_mul, sq, \u2190mul_assoc, mul_inv_cancel]; ring\n  apply _root_.ne_of_gt; apply selbergBoundingSum_pos;", "proofType": "term", "proofLengthLines": 16, "proofLengthTokens": 745}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\n", "theoremStatement": "lemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 ", "theoremName": "hh_integral'", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "7d53759dfc8d08e60e519bb2843d30571f54df85", "date": "2024-03-27"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1347, "tokenPositionInFile": 66105, "theoremPositionInFile": 119}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 143, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", 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"Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", 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+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module selberg\n-/\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic\n\n/-!\n# The Selberg Sieve\n\nThis file proves `selberg_bound_simple`, the main theorem of the Selberg.\n-/\n\nnoncomputable section\n\nopen scoped BigOperators Classical Sieve\n\nopen Finset Real Nat Sieve.UpperBoundSieve ArithmeticFunction Sieve\n\nstructure SelbergSieve extends Sieve where mk ::\n  level : \u211d\n  one_le_level : 1 \u2264 level\n\nnamespace SelbergSieve\nset_option quotPrecheck false\n\nvariable (s : SelbergSieve)\nlocal notation3 \"\u03bd\" => Sieve.nu (toSieve s)\nlocal notation3 \"P\" => Sieve.prodPrimes (toSieve s)\nlocal notation3 \"a\" => Sieve.weights (toSieve s)\nlocal notation3 \"X\" => Sieve.totalMass (toSieve s)\nlocal notation3 \"R\" => Sieve.rem (toSieve s)  -- this one seems broken\nlocal notation3 \"g\" => Sieve.selbergTerms (toSieve s)\nlocal notation3 \"y\" => SelbergSieve.level s\nlocal notation3 \"hy\" => SelbergSieve.one_le_level s\n\n--set_option profiler true\n@[simp]\ndef selbergBoundingSum : \u211d :=\n  \u2211 l in divisors P, if l ^ 2 \u2264 y then g l else 0\n\nset_option quotPrecheck false\nlocal notation3 \"S\" => SelbergSieve.selbergBoundingSum s\n\ntheorem selbergBoundingSum_pos :\n    0 < S := by\n  dsimp only [selbergBoundingSum]\n  rw [\u2190 sum_filter]\n  apply sum_pos;\n  \u00b7 intro l hl\n    rw [mem_filter, mem_divisors] at hl\n    \u00b7 apply s.selbergTerms_pos _ (hl.1.1)\n  \u00b7 simp_rw [Finset.Nonempty, mem_filter]; use 1\n    constructor\n    \u00b7 apply one_mem_divisors.mpr s.prodPrimes_ne_zero\n    rw [cast_one, one_pow]\n    exact s.one_le_level\n\ntheorem selbergBoundingSum_ne_zero : S \u2260 0 := by\n  apply _root_.ne_of_gt\n  exact s.selbergBoundingSum_pos\n\ntheorem selbergBoundingSum_nonneg : 0 \u2264 S := _root_.le_of_lt s.selbergBoundingSum_pos\n\ndef selbergWeights : \u2115 \u2192 \u211d := fun d =>\n  if d \u2223 P then\n    (\u03bd d)\u207b\u00b9 * g d * \u03bc d * S\u207b\u00b9 *\n      \u2211 m in divisors P, if (d * m) ^ 2 \u2264 y \u2227 m.Coprime d then g m else 0\n  else 0\n\n-- This notation traditionally uses \u03bb, which is unavailable in lean\nset_option quotPrecheck false\nlocal notation3 \"\u03b3\" => SelbergSieve.selbergWeights s\n\ntheorem selbergWeights_eq_zero_of_not_dvd {d : \u2115} (hd : \u00ac d \u2223 P) :\n    \u03b3 d = 0 := by\n  rw [selbergWeights, if_neg hd]\n\n", "theoremStatement": "theorem selbergWeights_eq_zero (d : \u2115) (hd : \u00acd ^ 2 \u2264 y) :\n    \u03b3 d = 0 ", "theoremName": "SelbergSieve.selbergWeights_eq_zero", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/Selberg.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", 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"Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  dsimp only [selbergWeights]\n  split_ifs with h\n  \u00b7 rw [mul_eq_zero_of_right _]\n    apply Finset.sum_eq_zero\n    refine fun m hm => if_neg ?_\n    intro hyp\n    have : (d^2:\u211d) \u2264 (d*m)^2 := by\n      norm_cast;\n      refine Nat.pow_le_pow_of_le_left ?h 2\n      exact Nat.le_mul_of_pos_right _ (Nat.pos_of_mem_divisors hm)\n    linarith [hyp.1]\n  \u00b7 rfl", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 354}}
+{"srcContext": "import Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Topology.Algebra.Order.Compact\nimport Mathlib.Analysis.SpecialFunctions.Log.Basic\n\nopen Filter Topology\n\nnamespace Asymptotics\n\nvariable {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*} {F : Type*} {G : Type*} {E' : Type*}\n  {F' : Type*} {G' : Type*} {E'' : Type*} {F'' : Type*} {G'' : Type*} {R : Type*}\n  {R' : Type*} {\ud835\udd5c : Type*} {\ud835\udd5c' : Type*}\n\nvariable [Norm E] [Norm F] [Norm G]\n\nvariable [SeminormedAddCommGroup E'] [SeminormedAddCommGroup F'] [SeminormedAddCommGroup G']\n  [NormedAddCommGroup E''] [NormedAddCommGroup F''] [NormedAddCommGroup G''] [SeminormedRing R]\n  [SeminormedRing R']\n\n\ntheorem isLittleO_const_id_cocompact [ProperSpace F'']\n    (c : E'') : (fun _x : F'' => c) =o[cocompact F''] id :=\n  isLittleO_const_left.2 <| Or.inr tendsto_norm_cocompact_atTop\n\n-- to replace existing `isLittleO_const_id_atTop`\ntheorem isLittleO_const_id_atTop2 [LinearOrder F''] [NoMaxOrder F''] [ClosedIciTopology F'']\n    [ProperSpace F''] (c : E'') : (fun _x : F'' => c) =o[atTop] id :=\n (isLittleO_const_id_cocompact c).mono atTop_le_cocompact\n\n-- to replace existing `isLittleO_const_id_atBot`\ntheorem isLittleO_const_id_atBot2 [LinearOrder F''] [NoMinOrder F''] [ClosedIicTopology F'']\n    [ProperSpace F''] (c : E'') : (fun _x : F'' => c) =o[atBot] id :=\n  (isLittleO_const_id_cocompact c).mono atBot_le_cocompact\n\ntheorem _root_.Filter.Eventually.natCast {f : \u211d \u2192 Prop} (hf : \u2200\u1da0 x in atTop, f x) : \u2200\u1da0 n : \u2115 in atTop, f n :=\n  tendsto_nat_cast_atTop_atTop.eventually hf\n\n", "theoremStatement": "theorem IsBigO.natCast {f g : \u211d \u2192 E} (h : f =O[atTop] g) :\n    (fun n : \u2115 => f n) =O[atTop] fun n : \u2115 => g n ", "theoremName": "Asymptotics.IsBigO.natCast", "fileCreated": {"commit": "84fbebeb60a5e72e2d8a4e6b350a46af7023f681", "date": "2024-02-08"}, "theoremCreated": {"commit": 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"Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", 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"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  h.comp_tendsto tendsto_nat_cast_atTop_atTop", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 48}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\n", "theoremStatement": "theorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x ", "theoremName": "sum_le_integral", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "3ede5cdd76d48e95a04ec9dba02689f6c6b7d956", "date": "2024-03-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1218, "tokenPositionInFile": 60893, "theoremPositionInFile": 115}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 431, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", 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"Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp", "proofType": "tactic", "proofLengthLines": 42, "proofLengthTokens": 1670}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\nlemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) := by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]\n\n/-%%\n\\begin{definition}[RiemannZeta0]\\label{RiemannZeta0}\\lean{RiemannZeta0}\\leanok\n\\uses{ZetaSum_aux2}\nFor any natural $N\\ge1$, we define\n$$\n\\zeta_0(N,s) :=\n\\sum_{1\\le n < N} \\frac1{n^s}\n+\n\\frac{- N^{1-s}}{1-s} + \\frac{-N^{-s}}{2} + s \\int_N^\\infty \\frac{\\lfloor x\\rfloor + 1/2 - x}{x^{s+1}} \\, dx\n$$\n\\end{definition}\n%%-/\nnoncomputable def riemannZeta0 (N : \u2115) (s : \u2102) : \u2102 :=\n  (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n  (- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) / (x : \u2102) ^ (s + 1)\n\n/-- We use `\u03b6` to denote the Rieman zeta function and `\u03b6\u2080` to denote the alternative\n  Rieman zeta function.. -/\nlocal notation (name := riemannzeta) \"\u03b6\" => riemannZeta\nlocal notation (name := riemannzeta0) \"\u03b6\u2080\" => riemannZeta0\n\nlemma riemannZeta0_apply (N : \u2115) (s : \u2102) : \u03b6\u2080 N s =\n    (\u2211 n in Finset.range N, 1 / (n : \u2102) ^ s) +\n    ((- N ^ (1 - s)) / (1 - s) + (- N ^ (-s)) / 2\n      + s * \u222b x in Ioi (N : \u211d), (\u230ax\u230b + 1 / 2 - x) * (x : \u2102) ^ (-(s + 1))) := by\n  simp_rw [riemannZeta0, div_cpow_eq_cpow_neg]; ring\n\n-- lemma AnalyticContinuation {f g : \u2102 \u2192 \u2102} {s t : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (f_eq_g_on_cap : EqOn f g (s \u2229 t))\n--     (s_open : IsOpen s) (t_open : IsOpen t) (cap_nonempty : Nonempty (s \u2229 t)) :\n--     \u2203! h : \u2102 \u2192 \u2102, AnalyticOn \u2102 h (s \u222a t) \u2227 EqOn h f s \u2227 EqOn h g t := by\n--   classical\n--   let h : \u2102 \u2192 \u2102 := fun z \u21a6 if z \u2208 s then f z else g z\n--   refine \u27e8h, \u27e8?_, fun z hz \u21a6 by simp [h, hz], ?_\u27e9, ?_\u27e9\n--   \u00b7 sorry\n--   \u00b7 intro z hz\n--     by_cases z_in_s : z \u2208 s\n--     \u00b7 have : z \u2208 s \u2229 t := by simp [z_in_s, hz]\n--       have := f_eq_g_on_cap this\n--       simp [h, z_in_s, this]\n--     \u00b7 simp [h, z_in_s]\n--   \u00b7 intro h' \u27e8h'_analytic, h'_eq_f_on_s, h'_eq_g_on_t\u27e9\n--     sorry\n\n-- lemma AnalyticContinuation' {f g : \u2102 \u2192 \u2102} {s t u : Set \u2102} (f_on_s : AnalyticOn \u2102 f s)\n--     (g_on_t : AnalyticOn \u2102 g t) (u_sub : u \u2286 s \u2229 t) (u_open : IsOpen u)\n--     (u_nonempty : Nonempty u) (f_eq_g_on_u : EqOn f g u) :\n--     EqOn f g (s \u2229 t) := by\n--   sorry\n\n-- move near `Real.differentiableAt_rpow_const_of_ne`\nlemma Real.differentiableAt_cpow_const_of_ne (s : \u2102) {x : \u211d} (xpos : 0 < x) :\n    DifferentiableAt \u211d (fun (x : \u211d) \u21a6 (x : \u2102) ^ s) x := by\n  apply DifferentiableAt.comp_ofReal (e := fun z \u21a6 z ^ s)\n  apply DifferentiableAt.cpow (by simp) (by simp) (by simp [xpos])\n\nlemma Complex.one_div_cpow_eq {s : \u2102} {x : \u211d} (x_ne : x \u2260 0) :\n    1 / (x : \u2102) ^ s = (x : \u2102) ^ (-s) := by\n  refine (eq_one_div_of_mul_eq_one_left ?_).symm\n  rw [\u2190 cpow_add _ _ <| mod_cast x_ne, add_left_neg, cpow_zero]\n\n-- No longer used\nlemma ContDiffOn.hasDeriv_deriv {\u03c6 : \u211d \u2192 \u2102} {s : Set \u211d} (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 s) {x : \u211d}\n    (x_in_s : s \u2208 nhds x) : HasDerivAt \u03c6 (deriv \u03c6 x) x :=\n  (ContDiffAt.hasStrictDerivAt (\u03c6Diff.contDiffAt x_in_s) (by simp)).hasDerivAt\n\n-- No longer used\nlemma ContDiffOn.continuousOn_deriv {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d}\n    (\u03c6Diff : ContDiffOn \u211d 1 \u03c6 (uIoo a b)) :\n    ContinuousOn (deriv \u03c6) (uIoo a b) := by\n  apply ContDiffOn.continuousOn (\ud835\udd5c := \u211d) (n := 0)\n  exact (fun h \u21a6 ((contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo).1 h).2) \u03c6Diff\n\nlemma LinearDerivative_ofReal (x : \u211d) (a b : \u2102) : HasDerivAt (fun (t : \u211d) \u21a6 a * t + b) a x := by\n  refine HasDerivAt.add_const ?_ b\n  convert (ContinuousLinearMap.hasDerivAt Complex.ofRealCLM).const_mul a using 1; simp\n-- No longer used\nsection\n-- from Floris van Doorn\n\nvariable {A : Type*} [NormedRing A] [NormedAlgebra \u211d A] [CompleteSpace A] {a b : \u211d}\n\nset_option autoImplicit false in\nopen BigOperators Interval Topology Set intervalIntegral MeasureTheory in\nlemma integral_deriv_mul_eq_sub' {u v u' v' : \u211d \u2192 A}\n    (hu : \u2200 x \u2208 [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)\n    (hv : \u2200 x \u2208 [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)\n    (hu' : IntervalIntegrable u' volume a b)\n    (hv' : IntervalIntegrable v' volume a b) :\n    \u222b x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a := by\n  have h2u : ContinuousOn u [[a, b]] := fun x hx \u21a6 (hu x hx).continuousWithinAt\n  have h2v : ContinuousOn v [[a, b]] := fun x hx \u21a6 (hv x hx).continuousWithinAt\n  apply integral_eq_sub_of_hasDeriv_right (h2u.mul h2v)\n  \u00b7 exact fun x hx \u21a6 (hu x <| mem_Icc_of_Ioo hx).mul (hv x <| mem_Icc_of_Ioo hx) |>.hasDerivAt\n      (Icc_mem_nhds hx.1 hx.2) |>.hasDerivWithinAt\n  \u00b7 exact (hu'.mul_continuousOn h2v).add (hv'.continuousOn_mul h2u)\n\nend\n\nlemma sum_eq_int_deriv_aux2 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} (c : \u2102)\n    (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u222b (x : \u211d) in a..b, (c - x) * deriv \u03c6 x =\n      (c - b) * \u03c6 b - (c - a) * \u03c6 a + \u222b (x : \u211d) in a..b, \u03c6 x := by\n  set u := fun (x : \u211d) \u21a6 c - x\n  set u' := fun (x : \u211d) \u21a6 (-1 : \u2102)\n  have hu : \u2200 x \u2208 uIcc a b, HasDerivAt u (u' x) x := by\n    exact fun x _ \u21a6 by convert LinearDerivative_ofReal x (-1 : \u2102) c; ring\n  have hu' : IntervalIntegrable u' MeasureTheory.volume a b := by\n    apply Continuous.intervalIntegrable; continuity\n  have hv' : IntervalIntegrable (deriv \u03c6) MeasureTheory.volume a b :=\n    deriv\u03c6Cont.intervalIntegrable\n  convert intervalIntegral.integral_mul_deriv_eq_deriv_mul hu \u03c6Diff hu' hv' using 1; simp [u]\n\nlemma sum_eq_int_deriv_aux_eq {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124}\n    (b_eq_kpOne : b = k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k + 1 := Int.floor_eq_iff.mpr \u27e8by exact_mod_cast b_eq_kpOne.symm.le,\n    by rw [b_eq_kpOne]; simp\u27e9\n  simp only [flb_eq_k, Finset.Icc_self, Finset.sum_singleton, Int.cast_add, Int.cast_one]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont, b_eq_kpOne]\n  ring_nf\n  have : Finset.Ioc k (1 + k) = {k + 1} := by\n    ext m\n    simp only [Finset.mem_Ioc, Finset.mem_singleton]\n    constructor\n    \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 by rw [add_comm] at h\u2082; exact Int.le_antisymm h\u2082 h\u2081\n    \u00b7 exact fun h \u21a6 \u27e8by simp [h], by simp [h, add_comm]\u27e9\n  simp_rw [this, Finset.sum_singleton, Int.cast_add, Int.cast_one, add_comm]\n\nlemma sum_eq_int_deriv_aux_lt {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_lt_kpOne : b < k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  have flb_eq_k : \u230ab\u230b = k := Int.floor_eq_iff.mpr \u27e8by linarith [ha.1, ha.2], by linarith\u27e9\n  simp only [flb_eq_k, gt_iff_lt, lt_add_iff_pos_right, zero_lt_one, Finset.Icc_eq_empty_of_lt,\n    Finset.sum_empty]\n  rw [sum_eq_int_deriv_aux2 (k + 1 / 2) \u03c6Diff deriv\u03c6Cont]\n  have : Finset.Ioc k k = {} := by simp only [ge_iff_le, le_refl, Finset.Ioc_eq_empty_of_le]\n  simp only [this, Finset.sum_empty, one_div]; ring_nf\n\nlemma sum_eq_int_deriv_aux1 {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc k \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (k + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (k + 1 / 2 - x) * deriv \u03c6 x := by\n  by_cases h : b = k + 1\n  \u00b7 exact sum_eq_int_deriv_aux_eq h \u03c6Diff deriv\u03c6Cont\n  \u00b7 exact sum_eq_int_deriv_aux_lt ha (Ne.lt_of_le h b_le_kpOne) \u03c6Diff deriv\u03c6Cont\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv_aux]\\label{sum_eq_int_deriv_aux}\\lean{sum_eq_int_deriv_aux}\\leanok\n  Let $k \\le a < b\\le k+1$, with $k$ an integer, and let $\\phi$ be continuously differentiable on\n  $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\nlemma sum_eq_int_deriv_aux {\u03c6 : \u211d \u2192 \u2102} {a b : \u211d} {k : \u2124} (ha : a \u2208 Ico (k : \u211d) b)\n    (b_le_kpOne : b \u2264 k + 1) (\u03c6Diff : \u2200 x \u2208 [[a, b]], HasDerivAt \u03c6 (deriv \u03c6 x) x)\n    (deriv\u03c6Cont : ContinuousOn (deriv \u03c6) [[a, b]]) :\n    \u2211 n in Finset.Ioc \u230aa\u230b \u230ab\u230b, \u03c6 n =\n    (\u222b x in a..b, \u03c6 x) + (\u230ab\u230b + 1 / 2 - b) * \u03c6 b - (\u230aa\u230b + 1 / 2 - a) * \u03c6 a\n      - \u222b x in a..b, (\u230ax\u230b + 1 / 2 - x) * deriv \u03c6 x := by\n  have fl_a_eq_k : \u230aa\u230b = k := Int.floor_eq_iff.mpr \u27e8ha.1, by linarith [ha.2]\u27e9\n  convert sum_eq_int_deriv_aux1 ha b_le_kpOne \u03c6Diff deriv\u03c6Cont using 2\n  \u00b7 rw [fl_a_eq_k]\n  \u00b7 congr\n  \u00b7 apply intervalIntegral.integral_congr_ae\n    have : \u2200\u1d50 (x : \u211d) \u2202MeasureTheory.volume, x \u2260 b := by\n      convert Countable.ae_not_mem (s := {b}) (by simp) (\u03bc := MeasureTheory.volume) using 1\n    filter_upwards [this]\n    intro x x_ne_b hx\n    rw [uIoc_of_le ha.2.le, mem_Ioc] at hx\n    congr\n    exact Int.floor_eq_iff.mpr \u27e8by linarith [ha.1], by have := Ne.lt_of_le x_ne_b hx.2; linarith\u27e9\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\n-- Thanks to Arend Mellendijk\n\nlemma interval_induction_aux_int (n : \u2115) : \u2200 (P : \u211d \u2192 \u211d \u2192 Prop)\n    (_ : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (_ : \u2200 (a : \u211d) (k : \u2124) (c : \u211d), a < k \u2192 k < c \u2192 P a k \u2192 P k c \u2192 P a c)\n    (a b : \u211d) (_ : a < b) (_ : n = \u230ab\u230b - \u230aa\u230b),\n    P a b := by\n  induction n using Nat.case_strong_induction_on with\n  | hz =>\n    intro P base _ a b hab hn\n    apply base a b \u230aa\u230b (Int.floor_le a) hab\n    rw [(by simp only [CharP.cast_eq_zero] at hn; linarith : \u230aa\u230b = \u230ab\u230b)]\n    exact (Int.lt_floor_add_one b).le\n  | hi n ih =>\n    intro P base step a b _ hn\n    have Pa : P a (\u230aa\u230b + 1) :=\n      base a (\u230aa\u230b + 1) \u230aa\u230b (Int.floor_le a) (Int.lt_floor_add_one a) (le_of_eq rfl)\n    by_cases b_le_flaP1 : b = \u230aa\u230b + 1\n    \u00b7 rwa [b_le_flaP1]\n    have flaP1_lt_b : \u230aa\u230b + 1 < b := by\n      simp only [Nat.cast_succ] at hn\n      have : (\u230aa\u230b : \u211d) + 1 \u2264 \u230ab\u230b := by exact_mod_cast (by linarith)\n      exact Ne.lt_of_le (id (Ne.symm b_le_flaP1)) (by linarith [Int.floor_le b] : \u230aa\u230b + 1 \u2264 b)\n    have Pfla_b : P (\u230aa\u230b + 1) b := by\n      apply ih n (le_of_eq rfl) P base step (\u230aa\u230b + 1) b flaP1_lt_b\n      simp only [Int.floor_add_one, Int.floor_intCast, Nat.cast_succ] at hn \u22a2\n      linarith\n    refine step a (\u230aa\u230b + 1) b ?_ (by exact_mod_cast flaP1_lt_b) (by exact_mod_cast Pa)\n      (by exact_mod_cast Pfla_b)\n    have := Int.lt_floor_add_one a\n    exact_mod_cast this\n\nlemma interval_induction (P : \u211d \u2192 \u211d \u2192 Prop)\n    (base : \u2200 a b : \u211d, \u2200 k : \u2124, k \u2264 a \u2192 a < b \u2192 b \u2264 k + 1 \u2192 P a b)\n    (step : \u2200 (a : \u211d) (k : \u2124) (b : \u211d), a < k \u2192 k < b \u2192 P a k \u2192 P k b \u2192 P a b)\n    (a b : \u211d) (hab : a < b) : P a b := by\n  set n := \u230ab\u230b - \u230aa\u230b with hn\n  clear_value n\n  have : 0 \u2264 n := by simp only [hn, sub_nonneg, ge_iff_le, Int.floor_le_floor _ _ (hab.le)]\n  lift n to \u2115 using this\n  exact interval_induction_aux_int n P base step a b hab hn\n\n/-%%\n\\begin{lemma}[sum_eq_int_deriv]\\label{sum_eq_int_deriv}\\lean{sum_eq_int_deriv}\\leanok\n  Let $a < b$, and let $\\phi$ be continuously differentiable on $[a, b]$.\n  Then\n  \\[\n  \\sum_{a < n \\le b} \\phi(n) = \\int_a^b \\phi(x) \\, dx + \\left(\\lfloor b \\rfloor + \\frac{1}{2} - b\\right) \\phi(b) - \\left(\\lfloor a \\rfloor + \\frac{1}{2} - a\\right) \\phi(a) - \\int_a^b \\left(\\lfloor x \\rfloor + \\frac{1}{2} - x\\right) \\phi'(x) \\, dx.\n  \\]\n\\end{lemma}\n%%-/\n/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {\u03b1 : Type*} [LinearOrder \u03b1] [LocallyFiniteOrder \u03b1]\n    {a b c: \u03b1} [DecidableEq \u03b1] (hb : b \u2208 Icc a c) : Ioc a b = Ioc a c \\ Ioc b c := by\n  ext x\n  simp only [mem_Ioc, mem_sdiff, not_and, not_le]\n  constructor\n  \u00b7 refine fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8\u27e8h\u2081, le_trans h\u2082 (mem_Icc.mp hb).2\u27e9, by contrapose! h\u2082; exact h\u2082.1\u27e9\n  \u00b7 exact fun \u27e8h\u2081, h\u2082\u27e9 \u21a6 \u27e8h\u2081.1, by contrapose! h\u2082; exact \u27e8h\u2082, h\u2081.2\u27e9\u27e9\n\n-- In Ya\u00ebl Dillies's API (https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Finset.2Esum_add_adjacent_intervals/near/430127101)\n", "theoremStatement": "lemma Finset.sum_Ioc_add_sum_Ioc {a b c : \u2124} (f : \u2124 \u2192 \u2102) (hb : b \u2208 Icc a c):\n    (\u2211 n in Finset.Ioc a b, f n) + (\u2211 n in Finset.Ioc b c, f n) = \u2211 n in Finset.Ioc a c, f n ", "theoremName": "Finset.sum_Ioc_add_sum_Ioc", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "8e5c683f6945de3190aa88ae5e78a1eff695c2a2", "date": "2024-03-28"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": "PrimeNumberTheoremAnd.ZetaBounds.jsonl", "positionMetadata": {"lineInFile": 289, "tokenPositionInFile": 13377, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, 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"Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  convert Finset.sum_sdiff (s\u2081 := Finset.Ioc b c) (s\u2082 := Finset.Ioc a c) ?_\n  \u00b7 exact Finset.Ioc_diff_Ioc hb\n  \u00b7 exact Finset.Ioc_subset_Ioc (mem_Icc.mp hb).1 (by rfl)", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 173}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\n", "theoremStatement": "lemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 ", "theoremName": "one_div_two_pi_mem_Ioo", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "556613ea25a3b2f304c446fcded05863f425c156", "date": "2024-03-25"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1152, "tokenPositionInFile": 57728, "theoremPositionInFile": 108}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 172, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 242}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]\n\nlemma Filter.BigO_zero_atTop_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[atTop] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioi_mem_atTop b] with c hc; replace hc := mem_Ioi.mp hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h]\n\n-- steal coerction lemmas from EulerProducts.Auxiliary because of build issues, and add new ones\nnamespace Complex\n-- see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Differentiability.20of.20the.20natural.20map.20.E2.84.9D.20.E2.86.92.20.E2.84.82/near/418095234\n\nlemma hasDerivAt_ofReal (x : \u211d) : HasDerivAt ofReal' 1 x :=\n  HasDerivAt.ofReal_comp <| hasDerivAt_id x\n\nlemma deriv_ofReal (x : \u211d) : deriv ofReal' x = 1 :=\n  (hasDerivAt_ofReal x).deriv\n\nlemma differentiableAt_ofReal (x : \u211d) : DifferentiableAt \u211d ofReal' x :=\n  (hasDerivAt_ofReal x).differentiableAt\n\nlemma differentiable_ofReal : Differentiable \u211d ofReal' :=\n  ofRealCLM.differentiable\n\nend Complex\n\nlemma DifferentiableAt.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    DifferentiableAt \u211d (fun x : \u211d \u21a6 e x) z :=\n  hf.hasDerivAt.comp_ofReal.differentiableAt\n\nlemma Differentiable.comp_ofReal {e : \u2102 \u2192 \u2102} (h : Differentiable \u2102 e) :\n    Differentiable \u211d (fun x : \u211d \u21a6 e x) :=\n  fun _ \u21a6 h.differentiableAt.comp_ofReal\n\nlemma DifferentiableAt.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} (hf : DifferentiableAt \u211d f z) :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z :=\n  hf.hasDerivAt.ofReal_comp.differentiableAt\n\nlemma Differentiable.ofReal_comp {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) :=\n  fun _ \u21a6 hf.differentiableAt.ofReal_comp\n\nopen Complex ContinuousLinearMap in\nlemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} {u : \u2102}\n    (hf : HasDerivAt (fun y \u21a6 (f y : \u2102)) u z) :\n    \u2203 u' : \u211d, u = u' \u2227 HasDerivAt f u' z := by\n  lift u to \u211d\n  \u00b7 have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv\n    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H\n    rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H\n  refine \u27e8u, rfl, ?_\u27e9\n  convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt\n  rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]\n\nlemma DifferentiableAt.ofReal_comp_iff {z : \u211d} {f : \u211d \u2192 \u211d} :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z \u2194 DifferentiableAt \u211d f z := by\n  refine \u27e8fun H \u21a6 ?_, ofReal_comp\u27e9\n  obtain \u27e8u, _, hu\u2082\u27e9 := H.hasDerivAt.of_hasDerivAt_ofReal_comp\n  exact HasDerivAt.differentiableAt hu\u2082\n\nlemma Differentiable.ofReal_comp_iff {f : \u211d \u2192 \u211d} :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) \u2194 Differentiable \u211d f :=\n  forall_congr' fun _ \u21a6 DifferentiableAt.ofReal_comp_iff\n\n", "theoremStatement": "lemma deriv.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} :\n    deriv (fun (y : \u211d) \u21a6 (f y : \u2102)) z = deriv f z ", "theoremName": "deriv.ofReal_comp", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "9fe9922a8d2b7d88dd6818ec9a6155204ba6d0a0", "date": "2024-03-29"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/MellinCalculus.lean", "module": "PrimeNumberTheoremAnd.MellinCalculus", "jsonFile": "PrimeNumberTheoremAnd.MellinCalculus.jsonl", "positionMetadata": {"lineInFile": 243, "tokenPositionInFile": 11120, 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"Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  by_cases hf : DifferentiableAt \u211d f z\n  \u00b7 exact hf.hasDerivAt.ofReal_comp.deriv\n  \u00b7 have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf\n    rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt <| hf',\n      Complex.ofReal_zero]", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 261}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\nlemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self\n\nlemma bound_sum_log0 {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let f0 i := if i = 0 then 0 else f i\n  have l1 : chebyWith C f0 := by\n    intro n ; refine Finset.sum_le_sum (fun i _ => ?_) |>.trans (hf n)\n    by_cases hi : i = 0 <;> simp [hi, f0]\n  have l2 i : \u2016f i\u2016 / i = \u2016f0 i\u2016 / i := by by_cases hi : i = 0 <;> simp [hi, f0]\n  simp_rw [l2] ; apply bound_sum_log rfl l1 hx\n\nlemma bound_sum_log' {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + 2 * \u03c0 ^ 2) := by\n  simpa only [hh_integral'] using bound_sum_log0 hf hx\n\nlemma summable_fourier (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  exact Summable.of_nonneg_of_le (fun _ => norm_nonneg _) l6 (by simpa using l5.const_smul (W21.norm \u03c8))\n\nlemma bound_I1 (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264\n    W21.norm \u03c8 \u2022 \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 := by\n\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  have l1 : Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n    exact summable_fourier x hx \u03c8 hcheby\n  apply (norm_tsum_le_tsum_norm l1).trans\n  simpa only [\u2190 tsum_const_smul _ l5] using tsum_mono l1 (by simpa using l5.const_smul (W21.norm \u03c8)) l6\n\nlemma bound_I1' {C : \u211d} (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21) (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264 W21.norm \u03c8 * C * (1 + 2 * \u03c0 ^ 2) := by\n\n  apply bound_I1 x (by linarith) \u03c8 \u27e8_, hcheby\u27e9 |>.trans\n  rw [smul_eq_mul, mul_assoc]\n  apply mul_le_mul le_rfl (bound_sum_log' hcheby hx) ?_ W21.norm_nonneg\n  apply tsum_nonneg (fun i => by positivity)\n\nlemma bound_I2 (x : \u211d) (\u03c8 : W21) :\n    \u2016\u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (2 * \u03c0 ^ 2) := by\n\n  have key a : \u2016\ud835\udcd5 \u03c8 (a / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := decay_bounds_key \u03c8 _\n  have twopi : 0 \u2264 2 * \u03c0 := by simp [pi_nonneg]\n  have l3 : Integrable (fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.comp_div (by norm_num [pi_ne_zero])\n  have l2 : IntegrableOn (fun i \u21a6 W21.norm \u03c8 * (1 + (i / (2 * \u03c0)) ^ 2)\u207b\u00b9) (Ici (-Real.log x)) := by\n    exact (l3.const_mul _).integrableOn\n  have l1 : IntegrableOn (fun i \u21a6 \u2016\ud835\udcd5 \u03c8 (i / (2 * \u03c0))\u2016) (Ici (-Real.log x)) := by\n    refine ((l3.const_mul (W21.norm \u03c8)).mono' ?_ ?_).integrableOn\n    \u00b7 apply Continuous.aestronglyMeasurable ; continuity\n    \u00b7 simp only [norm_norm, key] ; simp\n  have l5 : 0 \u2264\u1d50[volume] fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := by apply eventually_of_forall ; intro x ; positivity\n  refine (norm_integral_le_integral_norm _).trans <| (set_integral_mono l1 l2 key).trans ?_\n  rw [integral_mul_left] ; gcongr ; apply W21.norm_nonneg\n  refine (set_integral_le_integral l3 l5).trans ?_\n  rw [Measure.integral_comp_div (fun x => (1 + x ^ 2)\u207b\u00b9) (2 * \u03c0)]\n  simp [abs_eq_self.mpr twopi] ; ring_nf ; rfl\n\nlemma bound_main {C : \u211d} (A : \u2102) (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21)\n    (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264\n      W21.norm \u03c8 * (C * (1 + 2 * \u03c0 ^ 2) + \u2016A\u2016 * (2 * \u03c0 ^ 2)) := by\n\n  have l1 := bound_I1' x hx \u03c8 hcheby\n  have l2 := mul_le_mul (le_refl \u2016A\u2016) (bound_I2 x \u03c8) (by positivity) (by positivity)\n  apply norm_sub_le _ _ |>.trans ; rw [norm_mul]\n  convert _root_.add_le_add l1 l2 using 1 ; ring\n\n/-%%\n\\begin{lemma}[Limiting identity for Schwartz functions]\\label{schwarz-id}\\lean{limiting_cor_schwartz}\\leanok  The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.\n\\end{lemma}\n%%-/\n\nlemma limiting_cor_W21 (\u03c8 : W21) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) := by\n\n  -- Shorter notation for clarity\n  let S1 x (\u03c8 : \u211d \u2192 \u2102) := \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191n / x))\n  let S2 x (\u03c8 : \u211d \u2192 \u2102) := \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\n  let S x \u03c8 := S1 x \u03c8 - S2 x \u03c8 ; change Tendsto (fun x \u21a6 S x \u03c8) atTop (\ud835\udcdd 0)\n\n  -- Build the truncation\n  obtain g := exists_trunc\n  let \u03a8 R := g.scale R * \u03c8\n  have key R : Tendsto (fun x \u21a6 S x (\u03a8 R)) atTop (\ud835\udcdd 0) := limiting_cor (\u03a8 R) hf hcheby hG hG'\n\n  -- Choose the truncation radius\n  obtain \u27e8C, hcheby\u27e9 := hcheby\n  have hC : 0 \u2264 C := by\n    have : \u2016f 0\u2016 \u2264 C := by simpa [cumsum] using hcheby 1\n    have : 0 \u2264 \u2016f 0\u2016 := by positivity\n    linarith\n  have key2 : Tendsto (fun R \u21a6 W21.norm (\u03c8 - \u03a8 R)) atTop (\ud835\udcdd 0) := W21_approximation \u03c8 g\n  simp_rw [Metric.tendsto_nhds] at key key2 \u22a2 ; intro \u03b5 h\u03b5\n  let M := C * (1 + 2 * \u03c0 ^ 2) + \u2016(A : \u2102)\u2016 * (2 * \u03c0 ^ 2)\n  obtain \u27e8R, hR\u03c8\u27e9 := (key2 ((\u03b5 / 2) / (1 + M)) (by positivity)).exists\n  simp only [dist_zero_right, Real.norm_eq_abs, abs_eq_self.mpr W21.norm_nonneg] at hR\u03c8 key\n\n  -- Apply the compact support case\n  filter_upwards [eventually_ge_atTop 1, key R (\u03b5 / 2) (by positivity)] with x hx key\n\n  -- Control the tail term\n  have key3 : \u2016S x (\u03c8 - \u03a8 R)\u2016 < \u03b5 / 2 := by\n    have : \u2016S x _\u2016 \u2264 _ * M := @bound_main f C A x hx (\u03c8 - \u03a8 R) hcheby\n    apply this.trans_lt\n    apply (mul_le_mul (d := 1 + M) le_rfl (by simp) (by positivity) W21.norm_nonneg).trans_lt\n    have : 0 < 1 + M := by positivity\n    convert (mul_lt_mul_right this).mpr hR\u03c8 using 1 ; field_simp ; ring\n\n  -- Conclude the proof\n  have S1_sub_1 x : \ud835\udcd5 (\u21d1\u03c8 - \u21d1(\u03a8 R)) x = \ud835\udcd5 \u03c8 x - \ud835\udcd5 (\u03a8 R) x := by\n    have l1 : AEStronglyMeasurable (fun x_1 : \u211d \u21a6 cexp (-(2 * \u2191\u03c0 * (\u2191x_1 * \u2191x) * I))) volume := by\n      refine (Continuous.mul ?_ continuous_const).neg.cexp.aestronglyMeasurable\n      apply continuous_const.mul <| contDiff_ofReal.continuous.mul continuous_const\n    simp [Real.fourierIntegral_eq', mul_sub] ; apply integral_sub\n    \u00b7 apply \u03c8.hf.bdd_mul l1 ; use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n    \u00b7 apply (\u03a8 R : W21) |>.hf |>.bdd_mul l1\n      use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n\n  have S1_sub : S1 x (\u03c8 - \u03a8 R) = S1 x \u03c8 - S1 x (\u03a8 R) := by\n    simp [S1, S1_sub_1, mul_sub] ; apply tsum_sub\n    \u00b7 have := summable_fourier x (by positivity) \u03c8 \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n    \u00b7 have := summable_fourier x (by positivity) (\u03a8 R) \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n\n  have S2_sub : S2 x (\u03c8 - \u03a8 R) = S2 x \u03c8 - S2 x (\u03a8 R) := by\n    simp [S2, S1_sub_1] ; rw [integral_sub] ; ring\n    \u00b7 exact \u03c8.integrable_fourier (by positivity) |>.restrict\n    \u00b7 exact (\u03a8 R : W21).integrable_fourier (by positivity) |>.restrict\n\n  have S_sub : S x (\u03c8 - \u03a8 R) = S x \u03c8 - S x (\u03a8 R) := by simp [S, S1_sub, S2_sub] ; ring\n  simpa [S_sub, \u03a8] using norm_add_le _ _ |>.trans_lt (_root_.add_lt_add key3 key)\n\nlemma limiting_cor_schwartz (\u03c8 : \ud835\udce2(\u211d, \u2102)) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) :=\n  limiting_cor_W21 \u03c8 hf hcheby hG hG'\n\n/-%%\n\\begin{proof}\n\\uses{limiting-cor, smooth-ury}\\leanok\nFor any $R>1$, one can use a smooth cutoff function (provided by Lemma \\ref{smooth-ury} to write $\\psi = \\psi_{\\leq R} + \\psi_{>R}$, where $\\psi_{\\leq R}$ is $C^2$ (in fact smooth) and compactly supported (on $[-R,R]$), and $\\psi_{>R}$ obeys bounds of the form\n$$ |\\psi_{>R}(t)|, |\\psi''_{>R}(t)| \\ll R^{-1} / (1 + |t|^2) $$\nwhere the implied constants depend on $\\psi$.  By Lemma \\ref{decay} we then have\n$$ \\hat \\psi_{>R}(u) \\ll R^{-1} / (1+|u|^2).$$\nUsing this and \\eqref{cheby} one can show that\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{>R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ), A \\int_{-\\infty}^\\infty \\hat \\psi_{>R} (\\frac{u}{2\\pi})\\ du \\ll R^{-1} $$\n(with implied constants also depending on $A$), while from Lemma \\ref{limiting-cor} one has\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{\\leq R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi_{\\leq R} (\\frac{u}{2\\pi})\\ du + o(1).$$\nCombining the two estimates and letting $R$ be large, we obtain the claim.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Bijectivity of Fourier transform]\\label{bij}\\lean{fourier_surjection_on_schwartz}\\leanok  The Fourier transform is a bijection on the Schwartz class.\n\\end{lemma}\n%%-/\n\n-- just the surjectivity is stated here, as this is all that is needed for the current application, but perhaps one should state and prove bijectivity instead\n\nlemma fourier_surjection_on_schwartz (f : \ud835\udce2(\u211d, \u2102)) : \u2203 g : \ud835\udce2(\u211d, \u2102), \ud835\udcd5 g = f := by\n  use FS (FS (FS f)) ; ext x ; nth_rewrite 2 [\u2190 FS4 f] ; simp\n\n/-%%\n\\begin{proof}\n  \\leanok\n This is a standard result in Fourier analysis.\nIt can be proved here by appealing to Mellin inversion, Theorem \\ref{MellinInversion}.\nIn particular, given $f$ in the Schwartz class, let $F : \\R_+ \\to \\C : x \\mapsto f(\\log x)$ be a function in the ``Mellin space''; then the Mellin transform of $F$ on the imaginary axis $s=it$ is the Fourier transform of $f$.  The Mellin inversion theorem gives Fourier inversion.\n\\end{proof}\n%%-/\n\ndef toSchwartz (f : \u211d \u2192 \u2102) (h1 : ContDiff \u211d \u22a4 f) (h2 : HasCompactSupport f) : \ud835\udce2(\u211d, \u2102) where\n  toFun := f\n  smooth' := h1\n  decay' k n := by\n    have l1 : Continuous (fun x => \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := by\n      have : ContDiff \u211d \u22a4 (iteratedFDeriv \u211d n f) := h1.iteratedFDeriv_right le_top\n      exact Continuous.mul (by continuity) this.continuous.norm\n    have l2 : HasCompactSupport (fun x \u21a6 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := (h2.iteratedFDeriv _).norm.mul_left\n    simpa using l1.bounded_above_of_compact_support l2\n\n@[simp] lemma toSchwartz_apply (f : \u211d \u2192 \u2102) {h1 h2 x} : SchwartzMap.mk f h1 h2 x = f x := rfl\n\nlemma comp_exp_support0 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in \ud835\udcdd 0, \u03a8 x = 0 :=\n  not_mem_tsupport_iff_eventuallyEq.mp (fun h => lt_irrefl 0 <| mem_Ioi.mp (hplus h))\n\nlemma comp_exp_support1 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in atBot, \u03a8 (exp x) = 0 :=\n  Real.tendsto_exp_atBot <| comp_exp_support0 hplus\n\nlemma comp_exp_support2 {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) :\n    \u2200\u1da0 (x : \u211d) in atTop, (\u03a8 \u2218 rexp) x = 0 := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop] at hsupp\n  exact Real.tendsto_exp_atTop hsupp.2\n\ntheorem comp_exp_support {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    HasCompactSupport (\u03a8 \u2218 rexp) := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop]\n  exact \u27e8comp_exp_support1 hplus, comp_exp_support2 hsupp\u27e9\n\nlemma wiener_ikehara_smooth_aux (l0 : Continuous \u03a8) (hsupp : HasCompactSupport \u03a8)\n    (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) (x : \u211d) (hx : 0 < x) :\n    \u222b (u : \u211d) in Ioi (-Real.log x), \u2191(rexp u) * \u03a8 (rexp u) = \u222b (y : \u211d) in Ioi (1 / x), \u03a8 y := by\n\n  have l1 : ContinuousOn rexp (Ici (-Real.log x)) := by fun_prop\n  have l2 : Tendsto rexp atTop atTop := Real.tendsto_exp_atTop\n  have l3 t (_ : t \u2208 Ioi (-log x)) : HasDerivWithinAt rexp (rexp t) (Ioi t) t :=\n    (Real.hasDerivAt_exp t).hasDerivWithinAt\n  have l4 : ContinuousOn \u03a8 (rexp '' Ioi (-Real.log x)) := by fun_prop\n  have l5 : IntegrableOn \u03a8 (rexp '' Ici (-Real.log x)) volume :=\n    (l0.integrable_of_hasCompactSupport hsupp).integrableOn\n  have l6 : IntegrableOn (fun x \u21a6 rexp x \u2022 (\u03a8 \u2218 rexp) x) (Ici (-Real.log x)) volume := by\n    refine (Continuous.integrable_of_hasCompactSupport (by continuity) ?_).integrableOn\n    change HasCompactSupport (rexp \u2022 (\u03a8 \u2218 rexp))\n    exact (comp_exp_support hsupp hplus).smul_left\n  have := MeasureTheory.integral_comp_smul_deriv_Ioi l1 l2 l3 l4 l5 l6\n  simpa [Real.exp_neg, Real.exp_log hx] using this\n\ntheorem wiener_ikehara_smooth_sub (h1 : Integrable \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    Tendsto (fun x \u21a6 (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n\n  obtain \u27e8\u03b5, h\u03b5, hh\u27e9 := Metric.eventually_nhds_iff.mp <| comp_exp_support0 hplus\n  apply tendsto_nhds_of_eventually_eq ; filter_upwards [eventually_gt_atTop \u03b5\u207b\u00b9] with x hx\u03b5\n\n  have l1 : Integrable (indicator (Ioi x\u207b\u00b9) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n  have l2 : Integrable (indicator (Ioi 0) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n\n  simp_rw [\u2190 MeasureTheory.integral_indicator measurableSet_Ioi, \u2190 mul_sub, \u2190 integral_sub l1 l2]\n  simp ; right ; apply MeasureTheory.integral_eq_zero_of_ae ; apply eventually_of_forall ; intro t ; simp\n\n  have h\u03b5' : 0 < \u03b5\u207b\u00b9 := by positivity\n  have hx : 0 < x := by linarith\n  have hx' : 0 < x\u207b\u00b9 := by positivity\n  have h\u03b5x : x\u207b\u00b9 < \u03b5 := by apply (inv_lt h\u03b5 hx).mp hx\u03b5\n\n  have l3 : Ioi 0 = Ioc 0 x\u207b\u00b9 \u222a Ioi x\u207b\u00b9 := by\n    ext t ; simp ; constructor <;> intro h\n    \u00b7 simp [h, le_or_lt]\n    \u00b7 cases h <;> linarith\n  have l4 : Disjoint (Ioc 0 x\u207b\u00b9) (Ioi x\u207b\u00b9) := by simp\n  have l5 := Set.indicator_union_of_disjoint l4 \u03a8\n  rw [l3, l5] ; ring_nf\n  by_cases ht : t \u2208 Ioc 0 x\u207b\u00b9 <;> simp [ht]\n  apply hh ; simp at ht \u22a2\n  have : |t| \u2264 x\u207b\u00b9 := by rw [abs_le] ; constructor <;> linarith\n  linarith\n\n/-%%\n\\begin{corollary}[Smoothed Wiener-Ikehara]\\label{WienerIkeharaSmooth}\\lean{wiener_ikehara_smooth}\\leanok\n  If $\\Psi: (0,\\infty) \\to \\C$ is smooth and compactly supported away from the origin, then, then\n$$ \\sum_{n=1}^\\infty f(n) \\Psi( \\frac{n}{x} ) = A x \\int_0^\\infty \\Psi(y)\\ dy + o(x)$$\nas $u \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma wiener_ikehara_smooth (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x - A * \u222b y in Set.Ioi 0, \u03a8 y) atTop (nhds 0) := by\n\n  let h (x : \u211d) : \u2102 := rexp (2 * \u03c0 * x) * \u03a8 (exp (2 * \u03c0 * x))\n  have h1 : ContDiff \u211d \u22a4 h := by\n    have : ContDiff \u211d \u22a4 (fun x : \u211d => (rexp (2 * \u03c0 * x))) := (contDiff_const.mul contDiff_id).exp\n    exact (contDiff_ofReal.comp this).mul (hsmooth.comp this)\n  have h2 : HasCompactSupport h := by\n    have : 2 * \u03c0 \u2260 0 := by simp [pi_ne_zero]\n    simpa using (comp_exp_support hsupp hplus).comp_smul this |>.mul_left\n  obtain \u27e8g, hg\u27e9 := fourier_surjection_on_schwartz (toSchwartz h h1 h2)\n\n  have why (x : \u211d) : 2 * \u03c0 * x / (2 * \u03c0) = x := by field_simp ; ring\n  have l1 {y} (hy : 0 < y) : y * \u03a8 y = \ud835\udcd5 g (1 / (2 * \u03c0) * Real.log y) := by\n    field_simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [Real.exp_log hy]\n\n  have key := limiting_cor_schwartz g hf hcheby hG hG'\n\n  have l2 : \u2200\u1da0 x in atTop, \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 (\u21d1g) (1 / (2 * \u03c0) * Real.log (\u2191n / x)) =\n      \u2211' (n : \u2115), f n * \u03a8 (\u2191n / x) / x := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr ; ext n\n    by_cases hn : n = 0 ; simp [hn, (comp_exp_support0 hplus).self_of_nhds]\n    rw [\u2190 l1 (by positivity)]\n    have : (n : \u2102) \u2260 0 := by simpa using hn\n    have : (x : \u2102) \u2260 0 := by simpa using hx.ne.symm\n    field_simp ; ring\n\n  have l3 : \u2200\u1da0 x in atTop, \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 (\u21d1g) (u / (2 * \u03c0)) =\n      \u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr 1 ; simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [MeasureTheory.integral_Ici_eq_integral_Ioi]\n    exact wiener_ikehara_smooth_aux hsmooth.continuous hsupp hplus x hx\n\n  have l4 : Tendsto (fun x => (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n    exact wiener_ikehara_smooth_sub (hsmooth.continuous.integrable_of_hasCompactSupport hsupp) hplus\n\n  simpa [tsum_div_const] using (key.congr' <| EventuallyEq.sub l2 l3) |>.add l4\n\n/-%%\n\\begin{proof}\n\\uses{bij,schwarz-id}\\leanok\n By Lemma \\ref{bij}, we can write\n$$ y \\Psi(y) = \\hat \\psi( \\frac{1}{2\\pi} \\log y )$$\nfor all $y>0$ and some Schwartz function $\\psi$.  Making this substitution, the claim is then equivalent after standard manipulations to\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\nand the claim follows from Lemma \\ref{schwarz-id}.\n\\end{proof}\n%%-/\n\nlemma wiener_ikehara_smooth' (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) :=\n  tendsto_sub_nhds_zero_iff.mp <| wiener_ikehara_smooth hf hcheby hG hG' hsmooth hsupp hplus\n\nlocal instance {E : Type*} : Coe (E \u2192 \u211d) (E \u2192 \u2102) := \u27e8fun f n => f n\u27e9\n\n@[norm_cast]\ntheorem set_integral_ofReal {f : \u211d \u2192 \u211d} {s : Set \u211d} : \u222b x in s, (f x : \u2102) = \u222b x in s, f x :=\n  integral_ofReal\n\nlemma wiener_ikehara_smooth_real {f : \u2115 \u2192 \u211d} {\u03a8 : \u211d \u2192 \u211d} (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) := by\n\n  let \u03a8' := ofReal' \u2218 \u03a8\n  have l1 : ContDiff \u211d \u22a4 \u03a8' := contDiff_ofReal.comp hsmooth\n  have l2 : HasCompactSupport \u03a8' := hsupp.comp_left rfl\n  have l3 : closure (Function.support \u03a8') \u2286 Ioi 0 := by rwa [Function.support_comp_eq] ; simp\n  have key := (continuous_re.tendsto _).comp (@wiener_ikehara_smooth' A \u03a8 G f hf hcheby hG hG' l1 l2 l3)\n  simp at key ; norm_cast at key\n\nlemma interval_approx_inf (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      \u03c8 \u2264 indicator (Ico a b) 1 \u2227 b - a - \u03b5 \u2264 \u222b y in Ioi 0, \u03c8 y := by\n\n  have l1 : Iio ((b - a) / 3) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < (b - a) / 3)\n  have l2 : a < a + \u03b5 / 2 := by linarith\n  have l3 : b - \u03b5 / 2 < b := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 simp [h5, hab.ne, Icc_subset_Ioi_iff hab.le, ha]\n  \u00b7 exact h4.trans <| indicator_le_indicator_of_subset Ioo_subset_Ico_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a - \u03b5 := by linarith\n    have l5 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2229 Ioi 0 = Icc (a + \u03b5 / 2) (b - \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Icc (a + \u03b5 / 2) (b - \u03b5 / 2)) 1 y = b - a - \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; apply set_integral_mono ?_ l8 h3\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Icc]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma interval_approx_sup (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      indicator (Ico a b) 1 \u2264 \u03c8 \u2227 \u222b y in Ioi 0, \u03c8 y \u2264 b - a + \u03b5 := by\n\n  have l1 : Iio (a / 2) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < a / 2)\n  have l2 : a - \u03b5 / 2 < a := by linarith\n  have l3 : b < b + \u03b5 / 2 := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 have l4 : a - \u03b5 / 2 < b + \u03b5 / 2 := by linarith\n    have l5 : \u03b5 / 2 < a := by linarith\n    simp [h5, l4.ne, Icc_subset_Ioi_iff l4.le, l5]\n  \u00b7 apply le_trans ?_ h3\n    apply indicator_le_indicator_of_subset Ico_subset_Icc_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a + \u03b5 := by linarith\n    have l5 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2229 Ioi 0 = Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Ioo (a - \u03b5 / 2) (b + \u03b5 / 2)) 1 y = b - a + \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; refine set_integral_mono l8 ?_ h4\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Ioo]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma WI_summable {f : \u2115 \u2192 \u211d} {g : \u211d \u2192 \u211d} (hg : HasCompactSupport g) (hx : 0 < x) :\n    Summable (fun n => f n * g (n / x)) := by\n  obtain \u27e8M, hM\u27e9 := hg.bddAbove.mono subset_closure\n  apply summable_of_finite_support\n  simp ; apply Finite.inter_of_right ; rw [finite_iff_bddAbove]\n  exact \u27e8Nat.ceil (M * x), fun i hi => by simpa using Nat.ceil_mono ((div_le_iff hx).mp (hM hi))\u27e9\n\nlemma WI_sum_le {f : \u2115 \u2192 \u211d} {g\u2081 g\u2082 : \u211d \u2192 \u211d} (hf : 0 \u2264 f) (hg : g\u2081 \u2264 g\u2082) (hx : 0 < x)\n    (hg\u2081 : HasCompactSupport g\u2081) (hg\u2082 : HasCompactSupport g\u2082) :\n    (\u2211' n, f n * g\u2081 (n / x)) / x \u2264 (\u2211' n, f n * g\u2082 (n / x)) / x := by\n  apply div_le_div_of_nonneg_right ?_ hx.le\n  exact tsum_le_tsum (fun n => mul_le_mul_of_nonneg_left (hg _) (hf _)) (WI_summable hg\u2081 hx) (WI_summable hg\u2082 hx)\n\nlemma WI_sum_Iab_le {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) {C : \u211d} (hcheby : chebyWith C f) (hb : 0 < b) (hxb : 2 / b < x) :\n    (\u2211' n, f n * indicator (Ico a b) 1 (n / x)) / x \u2264 C * 2 * b := by\n  have hb' : 0 < 2 / b := by positivity\n  have hx : 0 < x := by linarith\n  have hxb' : 2 < x * b := (div_lt_iff hb).mp hxb\n  have l1 (i : \u2115) (hi : i \u2209 Finset.range \u2308b * x\u2309\u208a) : f i * indicator (Ico a b) 1 (i / x) = 0 := by\n    simp at hi \u22a2 ; right ; rintro - ; rw [le_div_iff hx] ; linarith\n  have l2 (i : \u2115) (_ : i \u2208 Finset.range \u2308b * x\u2309\u208a) : f i * indicator (Ico a b) 1 (i / x) \u2264 |f i| := by\n    rw [abs_eq_self.mpr (hpos _)]\n    convert_to _ \u2264 f i * 1 ; ring\n    apply mul_le_mul_of_nonneg_left ?_ (hpos _)\n    by_cases hi : (i / x) \u2208 (Ico a b) <;> simp [hi]\n  rw [tsum_eq_sum l1, div_le_iff hx, mul_assoc, mul_assoc]\n  apply Finset.sum_le_sum l2 |>.trans\n  have := hcheby \u2308b * x\u2309\u208a ; simp at this ; apply this.trans\n  have : 0 \u2264 C := by have := hcheby 1 ; simp [cumsum] at this ; exact (abs_nonneg _).trans this\n  refine mul_le_mul_of_nonneg_left ?_ this\n  apply (Nat.ceil_lt_add_one (by positivity)).le.trans\n  linarith\n\nlemma WI_sum_Iab_le' {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) {C : \u211d} (hcheby : chebyWith C f) (hb : 0 < b) :\n    \u2200\u1da0 x : \u211d in atTop, (\u2211' n, f n * indicator (Ico a b) 1 (n / x)) / x \u2264 C * 2 * b := by\n  filter_upwards [eventually_gt_atTop (2 / b)] with x hx using WI_sum_Iab_le hpos hcheby hb hx\n\nlemma le_of_eventually_nhdsWithin {a b : \u211d} (h : \u2200\u1da0 c in \ud835\udcdd[>] b, a \u2264 c) : a \u2264 b := by\n  apply le_of_forall_lt' ; intro d hd\n  have key : \u2200\u1da0 c in \ud835\udcdd[>] b, c < d := by\n    apply eventually_of_mem (U := Iio d) ?_ (fun x hx => hx)\n    rw [mem_nhdsWithin]\n    refine \u27e8Iio d, isOpen_Iio, hd, inter_subset_left _ _\u27e9\n  obtain \u27e8x, h1, h2\u27e9 := (h.and key).exists\n  linarith\n\nlemma ge_of_eventually_nhdsWithin {a b : \u211d} (h : \u2200\u1da0 c in \ud835\udcdd[<] b, c \u2264 a) : b \u2264 a := by\n  apply le_of_forall_lt ; intro d hd\n  have key : \u2200\u1da0 c in \ud835\udcdd[<] b, c > d := by\n    apply eventually_of_mem (U := Ioi d) ?_ (fun x hx => hx)\n    rw [mem_nhdsWithin]\n    refine \u27e8Ioi d, isOpen_Ioi, hd, inter_subset_left _ _\u27e9\n  obtain \u27e8x, h1, h2\u27e9 := (h.and key).exists\n  linarith\n\nlemma WI_tendsto_aux (a b : \u211d) {A : \u211d} (hA : 0 < A) :\n    Tendsto (fun c => c / A - (b - a)) (\ud835\udcdd[>] (A * (b - a))) (\ud835\udcdd[>] 0) := by\n  rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n  intro \u03b5 h\u03b5\n  refine \u27e8A * \u03b5, by positivity, ?_\u27e9\n  intro x hx1 hx2\n  constructor\n  \u00b7 simpa [lt_div_iff' hA]\n  \u00b7 simp [Real.dist_eq] at hx2 \u22a2\n    have : |x / A - (b - a)| = |x - A * (b - a)| / A := by\n      rw [\u2190 abs_eq_self.mpr hA.le, \u2190 abs_div, abs_eq_self.mpr hA.le] ; congr ; field_simp\n    rwa [this, div_lt_iff' hA]\n\nlemma WI_tendsto_aux' (a b : \u211d) {A : \u211d} (hA : 0 < A) :\n    Tendsto (fun c => (b - a) - c / A) (\ud835\udcdd[<] (A * (b - a))) (\ud835\udcdd[>] 0) := by\n  rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n  intro \u03b5 h\u03b5\n  refine \u27e8A * \u03b5, by positivity, ?_\u27e9\n  intro x hx1 hx2\n  constructor\n  \u00b7 simpa [div_lt_iff' hA]\n  \u00b7 simp [Real.dist_eq] at hx2 \u22a2\n    have : |(b - a) - x / A| = |A * (b - a) - x| / A := by\n      rw [\u2190 abs_eq_self.mpr hA.le, \u2190 abs_div, abs_eq_self.mpr hA.le] ; congr ; field_simp ; ring\n    rwa [this, div_lt_iff' hA, \u2190 neg_sub, abs_neg]\n\ntheorem residue_nonneg {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f)\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm (fun n \u21a6 \u2191(f n)) \u03c3')) (hcheby : cheby fun n \u21a6 \u2191(f n))\n    (hG : ContinuousOn G {s | 1 \u2264 s.re}) (hG' : EqOn G (fun s \u21a6 LSeries (fun n \u21a6 \u2191(f n)) s - \u2191A / (s - 1)) {s | 1 < s.re}) : 0 \u2264 A := by\n  let S (g : \u211d \u2192 \u211d) (x : \u211d) := (\u2211' n, f n * g (n / x)) / x\n  have hSnonneg {g : \u211d \u2192 \u211d} (hg : 0 \u2264 g) : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S g x := by\n    filter_upwards [eventually_ge_atTop 0] with x hx\n    exact div_nonneg (tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))) hx\n  obtain \u27e8\u03b5, \u03c8, h1, h2, h3, h4, -\u27e9 := (interval_approx_sup zero_lt_one one_lt_two).exists\n  have key := @wiener_ikehara_smooth_real A G f \u03c8 hf hcheby hG hG' h1 h2 h3\n  have l2 : 0 \u2264 \u03c8 := by apply le_trans _ h4 ; apply indicator_nonneg ; simp\n  have l1 : \u2200\u1da0 x in atTop, 0 \u2264 S \u03c8 x := hSnonneg l2\n  have l3 : 0 \u2264 A * \u222b (y : \u211d) in Ioi 0, \u03c8 y := ge_of_tendsto key l1\n  have l4 : 0 < \u222b (y : \u211d) in Ioi 0, \u03c8 y := by\n    have r1 : 0 \u2264\u1d50[Measure.restrict volume (Ioi 0)] \u03c8 := eventually_of_forall l2\n    have r2 : IntegrableOn (fun y \u21a6 \u03c8 y) (Ioi 0) volume :=\n      (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    have r3 : Ico 1 2 \u2286 Function.support \u03c8 := by intro x hx ; have := h4 x ; simp [hx] at this \u22a2 ; linarith\n    have r4 : Ico 1 2 \u2286 Function.support \u03c8 \u2229 Ioi 0 := by\n      simp [r3] ; apply Ico_subset_Icc_self.trans ; rw [Icc_subset_Ioi_iff] <;> linarith\n    have r5 : 1 \u2264 volume ((Function.support fun y \u21a6 \u03c8 y) \u2229 Ioi 0) := by convert volume.mono r4 ; norm_num\n    simpa [set_integral_pos_iff_support_of_nonneg_ae r1 r2] using zero_lt_one.trans_le r5\n  have := div_nonneg l3 l4.le ; field_simp at this ; exact this\n\n/-%%\nNow we add the hypothesis that $f(n) \\geq 0$ for all $n$.\n\n\\begin{proposition}[Wiener-Ikehara in an interval]\n\\label{WienerIkeharaInterval}\\lean{WienerIkeharaInterval}\\leanok\n  For any closed interval $I \\subset (0,+\\infty)$, we have\n  $$ \\sum_{n=1}^\\infty f(n) 1_I( \\frac{n}{x} ) = A x |I|  + o(x).$$\n\\end{proposition}\n%%-/\n\nlemma WienerIkeharaInterval {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a \u2264 b) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * (indicator (Ico a b) 1 (n / x))) / x) atTop (nhds (A * (b - a))) := by\n\n  -- Take care of the trivial case `a = b`\n  by_cases hab : a = b ; simp [hab] ; replace hb : a < b := lt_of_le_of_ne hb hab ; clear hab\n\n  -- Notation to make the proof more readable\n  let S (g : \u211d \u2192 \u211d) (x : \u211d) :=  (\u2211' n, f n * g (n / x)) / x\n  have hSnonneg {g : \u211d \u2192 \u211d} (hg : 0 \u2264 g) : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S g x := by\n    filter_upwards [eventually_ge_atTop 0] with x hx\n    refine div_nonneg ?_ hx\n    refine tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))\n  have hA : 0 \u2264 A := residue_nonneg hpos hf hcheby hG hG'\n\n  -- A few facts about the indicator function of `Icc a b`\n  let Iab : \u211d \u2192 \u211d := indicator (Ico a b) 1\n  change Tendsto (S Iab) atTop (\ud835\udcdd (A * (b - a)))\n  have hIab : HasCompactSupport Iab := by simpa [Iab, HasCompactSupport, tsupport, hb.ne] using isCompact_Icc\n  have Iab_nonneg : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S Iab x := hSnonneg (indicator_nonneg (by simp))\n  have Iab2 : IsBoundedUnder (\u00b7 \u2264 \u00b7) atTop (S Iab) := by\n    obtain \u27e8C, hC\u27e9 := hcheby ; exact \u27e8C * 2 * b, WI_sum_Iab_le' hpos hC (by linarith)\u27e9\n  have Iab3 : IsBoundedUnder (\u00b7 \u2265 \u00b7) atTop (S Iab) := \u27e80, Iab_nonneg\u27e9\n  have Iab0 : IsCoboundedUnder (\u00b7 \u2265 \u00b7) atTop (S Iab) := Iab2.isCoboundedUnder_ge\n  have Iab1 : IsCoboundedUnder (\u00b7 \u2264 \u00b7) atTop (S Iab) := Iab3.isCoboundedUnder_le\n\n  -- Bound from above by a smooth function\n  have sup_le : limsup (S Iab) atTop \u2264 A * (b - a) := by\n    have l_sup : \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, limsup (S Iab) atTop \u2264 A * (b - a + \u03b5) := by\n      filter_upwards [interval_approx_sup ha hb] with \u03b5 \u27e8\u03c8, h1, h2, h3, h4, h6\u27e9\n      have l1 : Tendsto (S \u03c8) atTop _ := wiener_ikehara_smooth_real hf hcheby hG hG' h1 h2 h3\n      have l6 : S Iab \u2264\u1da0[atTop] S \u03c8 := by\n        filter_upwards [eventually_gt_atTop 0] with x hx using WI_sum_le hpos h4 hx hIab h2\n      have l5 : IsBoundedUnder (\u00b7 \u2264 \u00b7) atTop (S \u03c8) := l1.isBoundedUnder_le\n      have l3 : limsup (S Iab) atTop \u2264 limsup (S \u03c8) atTop := limsup_le_limsup l6 Iab1 l5\n      apply l3.trans ; rw [l1.limsup_eq] ; gcongr\n    cases' (eq_or_ne A 0) with h h ; simpa [h] using l_sup\n    apply le_of_eventually_nhdsWithin\n    have key : 0 < A := lt_of_le_of_ne hA h.symm\n    filter_upwards [WI_tendsto_aux a b key l_sup] with x hx\n    simp at hx ; convert hx ; field_simp ; ring\n\n  -- Bound from below by a smooth function\n  have le_inf : A * (b - a) \u2264 liminf (S Iab) atTop := by\n    have l_inf : \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, A * (b - a - \u03b5) \u2264 liminf (S Iab) atTop := by\n      filter_upwards [interval_approx_inf ha hb] with \u03b5 \u27e8\u03c8, h1, h2, h3, h5, h6\u27e9\n      have l1 : Tendsto (S \u03c8) atTop _ := wiener_ikehara_smooth_real hf hcheby hG hG' h1 h2 h3\n      have l2 : S \u03c8 \u2264\u1da0[atTop] S Iab := by\n        filter_upwards [eventually_gt_atTop 0] with x hx using WI_sum_le hpos h5 hx h2 hIab\n      have l4 : IsBoundedUnder (\u00b7 \u2265 \u00b7) atTop (S \u03c8) := l1.isBoundedUnder_ge\n      have l3 : liminf (S \u03c8) atTop \u2264 liminf (S Iab) atTop := liminf_le_liminf l2 l4 Iab0\n      apply le_trans ?_ l3 ; rw [l1.liminf_eq] ; gcongr\n    cases' (eq_or_ne A 0) with h h ; simpa [h] using l_inf\n    apply ge_of_eventually_nhdsWithin\n    have key : 0 < A := lt_of_le_of_ne hA h.symm\n    filter_upwards [WI_tendsto_aux' a b key l_inf] with x hx\n    simp at hx ; convert hx ; field_simp ; ring\n\n  -- Combine the two bounds\n  have : liminf (S Iab) atTop \u2264 limsup (S Iab) atTop := liminf_le_limsup Iab2 Iab3\n  refine tendsto_of_liminf_eq_limsup ?_ ?_ Iab2 Iab3 <;> linarith\n\n/-%%\n\\begin{proof}\n\\uses{smooth-ury, WienerIkeharaSmooth} \\leanok\n  Use Lemma \\ref{smooth-ury} to bound $1_I$ above and below by smooth compactly supported functions whose integral is close to the measure of $|I|$, and use the non-negativity of $f$.\n\\end{proof}\n%%-/\n\nlemma le_floor_mul_iff (hb : 0 \u2264 b) (hx : 0 < x) : n \u2264 \u230ab * x\u230b\u208a \u2194 n / x \u2264 b := by\n  rw [div_le_iff hx, Nat.le_floor_iff] ; positivity\n\nlemma lt_ceil_mul_iff (hx : 0 < x) : n < \u2308b * x\u2309\u208a \u2194 n / x < b := by\n  rw [div_lt_iff hx, Nat.lt_ceil]\n\n", "theoremStatement": "lemma ceil_mul_le_iff (hx : 0 < x) : \u2308a * x\u2309\u208a \u2264 n \u2194 a \u2264 n / x ", "theoremName": "ceil_mul_le_iff", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "c916285851f4e3c2d71f05a4dd6a4c466318d6eb", "date": "2024-04-05"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 2016, "tokenPositionInFile": 101832, "theoremPositionInFile": 156}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 48, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", 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"Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", 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"PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [le_div_iff hx, Nat.ceil_le]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 39}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module selberg\n-/\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic\n\n/-!\n# The Selberg Sieve\n\nThis file proves `selberg_bound_simple`, the main theorem of the Selberg.\n-/\n\nnoncomputable section\n\nopen scoped BigOperators Classical Sieve\n\nopen Finset Real Nat Sieve.UpperBoundSieve ArithmeticFunction Sieve\n\nstructure SelbergSieve extends Sieve where mk ::\n  level : \u211d\n  one_le_level : 1 \u2264 level\n\nnamespace SelbergSieve\nset_option quotPrecheck false\n\nvariable (s : SelbergSieve)\nlocal notation3 \"\u03bd\" => Sieve.nu (toSieve s)\nlocal notation3 \"P\" => Sieve.prodPrimes (toSieve s)\nlocal notation3 \"a\" => Sieve.weights (toSieve s)\nlocal notation3 \"X\" => Sieve.totalMass (toSieve s)\nlocal notation3 \"R\" => Sieve.rem (toSieve s)  -- this one seems broken\nlocal notation3 \"g\" => Sieve.selbergTerms (toSieve s)\nlocal notation3 \"y\" => SelbergSieve.level s\nlocal notation3 \"hy\" => SelbergSieve.one_le_level s\n\n--set_option profiler true\n@[simp]\ndef selbergBoundingSum : \u211d :=\n  \u2211 l in divisors P, if l ^ 2 \u2264 y then g l else 0\n\nset_option quotPrecheck false\nlocal notation3 \"S\" => SelbergSieve.selbergBoundingSum s\n\n", "theoremStatement": "theorem selbergBoundingSum_pos :\n    0 < S ", "theoremName": "SelbergSieve.selbergBoundingSum_pos", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/Selberg.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg.jsonl", "positionMetadata": {"lineInFile": 47, "tokenPositionInFile": 1308, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, 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"Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  dsimp only [selbergBoundingSum]\n  rw [\u2190 sum_filter]\n  apply sum_pos;\n  \u00b7 intro l hl\n    rw [mem_filter, mem_divisors] at hl\n    \u00b7 apply s.selbergTerms_pos _ (hl.1.1)\n  \u00b7 simp_rw [Finset.Nonempty, mem_filter]; use 1\n    constructor\n    \u00b7 apply one_mem_divisors.mpr s.prodPrimes_ne_zero\n    rw [cast_one, one_pow]\n    exact s.one_le_level", "proofType": "tactic", "proofLengthLines": 11, "proofLengthTokens": 344}}
+{"srcContext": "/-\nCopyright (c) 2024 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.Analysis.Calculus.LineDeriv.Basic\nimport Mathlib.MeasureTheory.Integral.IntegralEqImproper\n-- import Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts\n\n/-!\n# Integration by parts for line derivatives\n\nLet `f, g : E \u2192 \u211d` be two differentiable functions on a real vector space endowed with a Haar\nmeasure. Then `\u222b f * g' = - \u222b f' * g`, where `f'` and `g'` denote the derivatives of `f` and `g`\nin a given direction `v`, provided that `f * g`, `f' * g` and `f * g'` are all integrable.\n\nIn this file, we prove this theorem as well as more general versions where the multiplication is\nreplaced by a general continuous bilinear form, giving versions both for the line derivative and\nthe Fr\u00e9chet derivative. These results are derived from the one-dimensional version and a Fubini\nargument.\n\n## Main statements\n\n* `integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable`: integration by parts\n  in terms of line derivatives, with `HasLineDerivAt` assumptions and general bilinear form.\n* `integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable`: integration by parts\n  in terms of Fr\u00e9chet derivatives, with `HasFDerivAt` assumptions and general bilinear form.\n* `integral_bilinear_fderiv_right_eq_neg_left_of_integrable`: integration by parts\n  in terms of Fr\u00e9chet derivatives, written with `fderiv` assumptions and general bilinear form.\n* `integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable`: integration by parts for scalar\n  action, in terms of Fr\u00e9chet derivatives, written with `fderiv` assumptions.\n* `integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable`: integration by parts for scalar\n  multiplication, in terms of Fr\u00e9chet derivatives, written with `fderiv` assumptions.\n\n-/\n\nopen MeasureTheory Measure FiniteDimensional\n\nvariable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [NormedAddCommGroup F]\n  [NormedSpace \u211d F] [NormedAddCommGroup G] [NormedSpace \u211d G] [NormedAddCommGroup W]\n  [NormedSpace \u211d W] [MeasurableSpace E] [BorelSpace E] {\u03bc : Measure E}\n\nlemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 [SigmaFinite \u03bc]\n    {f f' : E \u00d7 \u211d \u2192 F} {g g' : E \u00d7 \u211d \u2192 G} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) (\u03bc.prod volume))\n    (hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) (\u03bc.prod volume))\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) (\u03bc.prod volume))\n    (hf : \u2200 x, HasLineDerivAt \u211d f (f' x) x (0, 1)) (hg : \u2200 x, HasLineDerivAt \u211d g (g' x) x (0, 1)) :\n    \u222b x, B (f x) (g' x) \u2202(\u03bc.prod volume) = - \u222b x, B (f' x) (g x) \u2202(\u03bc.prod volume) := calc\n  \u222b x, B (f x) (g' x) \u2202(\u03bc.prod volume)\n    = \u222b x, (\u222b t, B (f (x, t)) (g' (x, t))) \u2202\u03bc := integral_prod _ hfg'\n  _ = \u222b x, (- \u222b t, B (f' (x, t)) (g (x, t))) \u2202\u03bc := by\n    apply integral_congr_ae\n    filter_upwards [hf'g.prod_right_ae, hfg'.prod_right_ae, hfg.prod_right_ae]\n      with x hf'gx hfg'x  hfgx\n    apply integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable ?_ ?_ hfg'x hf'gx hfgx\n    \u00b7 intro t\n      convert (hf (x, t)).scomp_of_eq t ((hasDerivAt_id t).add (hasDerivAt_const t (-t))) (by simp)\n        <;> simp\n    \u00b7 intro t\n      convert (hg (x, t)).scomp_of_eq t ((hasDerivAt_id t).add (hasDerivAt_const t (-t))) (by simp)\n        <;> simp\n  _ = - \u222b x, B (f' x) (g x) \u2202(\u03bc.prod volume) := by rw [integral_neg, integral_prod _ hf'g]\n\nlemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2\n    [FiniteDimensional \u211d E] {\u03bc : Measure (E \u00d7 \u211d)} [IsAddHaarMeasure \u03bc]\n    {f f' : E \u00d7 \u211d \u2192 F} {g g' : E \u00d7 \u211d \u2192 G} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) \u03bc)\n    (hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) \u03bc)\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) \u03bc)\n    (hf : \u2200 x, HasLineDerivAt \u211d f (f' x) x (0, 1)) (hg : \u2200 x, HasLineDerivAt \u211d g (g' x) x (0, 1)) :\n    \u222b x, B (f x) (g' x) \u2202\u03bc = - \u222b x, B (f' x) (g x) \u2202\u03bc := by\n  let \u03bd : Measure E := addHaar\n  have A : \u03bd.prod volume = (addHaarScalarFactor (\u03bd.prod volume) \u03bc) \u2022 \u03bc :=\n    isAddLeftInvariant_eq_smul _ _\n  have Hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) (\u03bd.prod volume) := by\n    rw [A]; exact hf'g.smul_measure_nnreal\n  have Hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) (\u03bd.prod volume) := by\n    rw [A]; exact hfg'.smul_measure_nnreal\n  have Hfg : Integrable (fun x \u21a6 B (f x) (g x)) (\u03bd.prod volume) := by\n    rw [A]; exact hfg.smul_measure_nnreal\n  rw [isAddLeftInvariant_eq_smul \u03bc (\u03bd.prod volume)]\n  simp [integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 Hf'g Hfg' Hfg hf hg]\n\nvariable [FiniteDimensional \u211d E] [IsAddHaarMeasure \u03bc]\n\n/-- **Integration by parts for line derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives\nof `f` and `g` in a given direction `v`, then `\u222b B f g' = - \u222b B f' g`. -/\ntheorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable\n    {f f' : E \u2192 F} {g g' : E \u2192 G} {v : E} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x) (g x)) \u03bc) (hfg' : Integrable (fun x \u21a6 B (f x) (g' x)) \u03bc)\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) \u03bc)\n    (hf : \u2200 x, HasLineDerivAt \u211d f (f' x) x v) (hg : \u2200 x, HasLineDerivAt \u211d g (g' x) x v) :\n    \u222b x, B (f x) (g' x) \u2202\u03bc = - \u222b x, B (f' x) (g x) \u2202\u03bc := by\n  by_cases hW : CompleteSpace W; swap\n  \u00b7 simp [integral, hW]\n  rcases eq_or_ne v 0 with rfl|hv\n  \u00b7 have Hf' x : f' x = 0 := by\n      simpa [(hasLineDerivAt_zero (f := f) (x := x)).lineDeriv] using (hf x).lineDeriv.symm\n    have Hg' x : g' x = 0 := by\n      simpa [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv] using (hg x).lineDeriv.symm\n    simp [Hf', Hg']\n  have : Nontrivial E := nontrivial_iff.2 \u27e8v, 0, hv\u27e9\n  let n := finrank \u211d E\n  let E' := Fin (n - 1) \u2192 \u211d\n  obtain \u27e8L, hL\u27e9 : \u2203 L : E \u2243L[\u211d] (E' \u00d7 \u211d), L v = (0, 1) := by\n    have : finrank \u211d (E' \u00d7 \u211d) = n := by simpa [this, E'] using Nat.sub_add_cancel finrank_pos\n    have L\u2080 : E \u2243L[\u211d] (E' \u00d7 \u211d) := (ContinuousLinearEquiv.ofFinrankEq this).symm\n    obtain \u27e8M, hM\u27e9 : \u2203 M : (E' \u00d7 \u211d) \u2243L[\u211d] (E' \u00d7 \u211d), M (L\u2080 v) = (0, 1) := by\n      apply SeparatingDual.exists_continuousLinearEquiv_apply_eq\n      \u00b7 simpa using hv\n      \u00b7 simp\n    exact \u27e8L\u2080.trans M, by simp [hM]\u27e9\n  let \u03bd := Measure.map L \u03bc\n  suffices H : \u222b (x : E' \u00d7 \u211d), (B (f (L.symm x))) (g' (L.symm x)) \u2202\u03bd =\n      -\u222b (x : E' \u00d7 \u211d), (B (f' (L.symm x))) (g (L.symm x)) \u2202\u03bd by\n    have : \u03bc = Measure.map L.symm \u03bd := by\n      simp [Measure.map_map L.symm.continuous.measurable L.continuous.measurable]\n    have hL : ClosedEmbedding L.symm := L.symm.toHomeomorph.closedEmbedding\n    simpa [this, hL.integral_map] using H\n  have L_emb : MeasurableEmbedding L := L.toHomeomorph.measurableEmbedding\n  apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2\n  \u00b7 simpa [L_emb.integrable_map_iff, Function.comp] using hf'g\n  \u00b7 simpa [L_emb.integrable_map_iff, Function.comp] using hfg'\n  \u00b7 simpa [L_emb.integrable_map_iff, Function.comp] using hfg\n  \u00b7 intro x\n    have : f = (f \u2218 L.symm) \u2218 (L : E \u2192\u2097[\u211d] (E' \u00d7 \u211d)) := by ext y; simp\n    specialize hf (L.symm x)\n    rw [this] at hf\n    convert hf.of_comp using 1\n    \u00b7 simp\n    \u00b7 simp [\u2190 hL]\n  \u00b7 intro x\n    have : g = (g \u2218 L.symm) \u2218 (L : E \u2192\u2097[\u211d] (E' \u00d7 \u211d)) := by ext y; simp\n    specialize hg (L.symm x)\n    rw [this] at hg\n    convert hg.of_comp using 1\n    \u00b7 simp\n    \u00b7 simp [\u2190 hL]\n\n/-- **Integration by parts for Fr\u00e9chet derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are derivatives\nof `f` and `g` in a given direction `v`, then `\u222b B f g' = - \u222b B f' g`. -/\ntheorem integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable\n    {f : E \u2192 F} {f' : E \u2192 (E \u2192L[\u211d] F)}\n    {g : E \u2192 G} {g' : E \u2192 (E \u2192L[\u211d] G)} {v : E} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (f' x v) (g x)) \u03bc)\n    (hfg' : Integrable (fun x \u21a6 B (f x) (g' x v)) \u03bc)\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) \u03bc)\n    (hf : \u2200 x, HasFDerivAt f (f' x) x) (hg : \u2200 x, HasFDerivAt g (g' x) x) :\n    \u222b x, B (f x) (g' x v) \u2202\u03bc = - \u222b x, B (f' x v) (g x) \u2202\u03bc :=\n  integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable hf'g hfg' hfg\n    (fun x \u21a6 (hf x).hasLineDerivAt v) (fun x \u21a6 (hg x).hasLineDerivAt v)\n\n", "theoremStatement": "/-- **Integration by parts for Fr\u00e9chet derivatives**\nVersion with a general bilinear form `B`.\nIf `B f g` is integrable, as well as `B f' g` and `B f g'` where `f'` and `g'` are the derivatives\nof `f` and `g` in a given direction `v`, then `\u222b B f g' = - \u222b B f' g`. -/\ntheorem integral_bilinear_fderiv_right_eq_neg_left_of_integrable\n    {f : E \u2192 F} {g : E \u2192 G} {v : E} {B : F \u2192L[\u211d] G \u2192L[\u211d] W}\n    (hf'g : Integrable (fun x \u21a6 B (fderiv \u211d f x v) (g x)) \u03bc)\n    (hfg' : Integrable (fun x \u21a6 B (f x) (fderiv \u211d g x v)) \u03bc)\n    (hfg : Integrable (fun x \u21a6 B (f x) (g x)) \u03bc)\n    (hf : Differentiable \u211d f) (hg : Differentiable \u211d g) :\n    \u222b x, B (f x) (fderiv \u211d g x v) \u2202\u03bc = - \u222b x, B (fderiv \u211d f x v) (g x) \u2202\u03bc ", "theoremName": "integral_bilinear_fderiv_right_eq_neg_left_of_integrable", "fileCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "theoremCreated": {"commit": "0138a287f668f7bb1c6b3f1009f55d9acd124dc3", "date": "2024-04-07"}, "file": 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"Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", 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"Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable hf'g hfg' hfg\n    (fun x \u21a6 (hf x).hasFDerivAt) (fun x \u21a6 (hg x).hasFDerivAt)", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 142}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\nlemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self\n\nlemma bound_sum_log0 {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let f0 i := if i = 0 then 0 else f i\n  have l1 : chebyWith C f0 := by\n    intro n ; refine Finset.sum_le_sum (fun i _ => ?_) |>.trans (hf n)\n    by_cases hi : i = 0 <;> simp [hi, f0]\n  have l2 i : \u2016f i\u2016 / i = \u2016f0 i\u2016 / i := by by_cases hi : i = 0 <;> simp [hi, f0]\n  simp_rw [l2] ; apply bound_sum_log rfl l1 hx\n\nlemma bound_sum_log' {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + 2 * \u03c0 ^ 2) := by\n  simpa only [hh_integral'] using bound_sum_log0 hf hx\n\nlemma summable_fourier (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  exact Summable.of_nonneg_of_le (fun _ => norm_nonneg _) l6 (by simpa using l5.const_smul (W21.norm \u03c8))\n\nlemma bound_I1 (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264\n    W21.norm \u03c8 \u2022 \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 := by\n\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  have l1 : Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n    exact summable_fourier x hx \u03c8 hcheby\n  apply (norm_tsum_le_tsum_norm l1).trans\n  simpa only [\u2190 tsum_const_smul _ l5] using tsum_mono l1 (by simpa using l5.const_smul (W21.norm \u03c8)) l6\n\nlemma bound_I1' {C : \u211d} (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21) (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264 W21.norm \u03c8 * C * (1 + 2 * \u03c0 ^ 2) := by\n\n  apply bound_I1 x (by linarith) \u03c8 \u27e8_, hcheby\u27e9 |>.trans\n  rw [smul_eq_mul, mul_assoc]\n  apply mul_le_mul le_rfl (bound_sum_log' hcheby hx) ?_ W21.norm_nonneg\n  apply tsum_nonneg (fun i => by positivity)\n\nlemma bound_I2 (x : \u211d) (\u03c8 : W21) :\n    \u2016\u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (2 * \u03c0 ^ 2) := by\n\n  have key a : \u2016\ud835\udcd5 \u03c8 (a / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := decay_bounds_key \u03c8 _\n  have twopi : 0 \u2264 2 * \u03c0 := by simp [pi_nonneg]\n  have l3 : Integrable (fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.comp_div (by norm_num [pi_ne_zero])\n  have l2 : IntegrableOn (fun i \u21a6 W21.norm \u03c8 * (1 + (i / (2 * \u03c0)) ^ 2)\u207b\u00b9) (Ici (-Real.log x)) := by\n    exact (l3.const_mul _).integrableOn\n  have l1 : IntegrableOn (fun i \u21a6 \u2016\ud835\udcd5 \u03c8 (i / (2 * \u03c0))\u2016) (Ici (-Real.log x)) := by\n    refine ((l3.const_mul (W21.norm \u03c8)).mono' ?_ ?_).integrableOn\n    \u00b7 apply Continuous.aestronglyMeasurable ; continuity\n    \u00b7 simp only [norm_norm, key] ; simp\n  have l5 : 0 \u2264\u1d50[volume] fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := by apply eventually_of_forall ; intro x ; positivity\n  refine (norm_integral_le_integral_norm _).trans <| (set_integral_mono l1 l2 key).trans ?_\n  rw [integral_mul_left] ; gcongr ; apply W21.norm_nonneg\n  refine (set_integral_le_integral l3 l5).trans ?_\n  rw [Measure.integral_comp_div (fun x => (1 + x ^ 2)\u207b\u00b9) (2 * \u03c0)]\n  simp [abs_eq_self.mpr twopi] ; ring_nf ; rfl\n\nlemma bound_main {C : \u211d} (A : \u2102) (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21)\n    (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264\n      W21.norm \u03c8 * (C * (1 + 2 * \u03c0 ^ 2) + \u2016A\u2016 * (2 * \u03c0 ^ 2)) := by\n\n  have l1 := bound_I1' x hx \u03c8 hcheby\n  have l2 := mul_le_mul (le_refl \u2016A\u2016) (bound_I2 x \u03c8) (by positivity) (by positivity)\n  apply norm_sub_le _ _ |>.trans ; rw [norm_mul]\n  convert _root_.add_le_add l1 l2 using 1 ; ring\n\n/-%%\n\\begin{lemma}[Limiting identity for Schwartz functions]\\label{schwarz-id}\\lean{limiting_cor_schwartz}\\leanok  The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.\n\\end{lemma}\n%%-/\n\nlemma limiting_cor_W21 (\u03c8 : W21) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) := by\n\n  -- Shorter notation for clarity\n  let S1 x (\u03c8 : \u211d \u2192 \u2102) := \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191n / x))\n  let S2 x (\u03c8 : \u211d \u2192 \u2102) := \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\n  let S x \u03c8 := S1 x \u03c8 - S2 x \u03c8 ; change Tendsto (fun x \u21a6 S x \u03c8) atTop (\ud835\udcdd 0)\n\n  -- Build the truncation\n  obtain g := exists_trunc\n  let \u03a8 R := g.scale R * \u03c8\n  have key R : Tendsto (fun x \u21a6 S x (\u03a8 R)) atTop (\ud835\udcdd 0) := limiting_cor (\u03a8 R) hf hcheby hG hG'\n\n  -- Choose the truncation radius\n  obtain \u27e8C, hcheby\u27e9 := hcheby\n  have hC : 0 \u2264 C := by\n    have : \u2016f 0\u2016 \u2264 C := by simpa [cumsum] using hcheby 1\n    have : 0 \u2264 \u2016f 0\u2016 := by positivity\n    linarith\n  have key2 : Tendsto (fun R \u21a6 W21.norm (\u03c8 - \u03a8 R)) atTop (\ud835\udcdd 0) := W21_approximation \u03c8 g\n  simp_rw [Metric.tendsto_nhds] at key key2 \u22a2 ; intro \u03b5 h\u03b5\n  let M := C * (1 + 2 * \u03c0 ^ 2) + \u2016(A : \u2102)\u2016 * (2 * \u03c0 ^ 2)\n  obtain \u27e8R, hR\u03c8\u27e9 := (key2 ((\u03b5 / 2) / (1 + M)) (by positivity)).exists\n  simp only [dist_zero_right, Real.norm_eq_abs, abs_eq_self.mpr W21.norm_nonneg] at hR\u03c8 key\n\n  -- Apply the compact support case\n  filter_upwards [eventually_ge_atTop 1, key R (\u03b5 / 2) (by positivity)] with x hx key\n\n  -- Control the tail term\n  have key3 : \u2016S x (\u03c8 - \u03a8 R)\u2016 < \u03b5 / 2 := by\n    have : \u2016S x _\u2016 \u2264 _ * M := @bound_main f C A x hx (\u03c8 - \u03a8 R) hcheby\n    apply this.trans_lt\n    apply (mul_le_mul (d := 1 + M) le_rfl (by simp) (by positivity) W21.norm_nonneg).trans_lt\n    have : 0 < 1 + M := by positivity\n    convert (mul_lt_mul_right this).mpr hR\u03c8 using 1 ; field_simp ; ring\n\n  -- Conclude the proof\n  have S1_sub_1 x : \ud835\udcd5 (\u21d1\u03c8 - \u21d1(\u03a8 R)) x = \ud835\udcd5 \u03c8 x - \ud835\udcd5 (\u03a8 R) x := by\n    have l1 : AEStronglyMeasurable (fun x_1 : \u211d \u21a6 cexp (-(2 * \u2191\u03c0 * (\u2191x_1 * \u2191x) * I))) volume := by\n      refine (Continuous.mul ?_ continuous_const).neg.cexp.aestronglyMeasurable\n      apply continuous_const.mul <| contDiff_ofReal.continuous.mul continuous_const\n    simp [Real.fourierIntegral_eq', mul_sub] ; apply integral_sub\n    \u00b7 apply \u03c8.hf.bdd_mul l1 ; use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n    \u00b7 apply (\u03a8 R : W21) |>.hf |>.bdd_mul l1\n      use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n\n  have S1_sub : S1 x (\u03c8 - \u03a8 R) = S1 x \u03c8 - S1 x (\u03a8 R) := by\n    simp [S1, S1_sub_1, mul_sub] ; apply tsum_sub\n    \u00b7 have := summable_fourier x (by positivity) \u03c8 \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n    \u00b7 have := summable_fourier x (by positivity) (\u03a8 R) \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n\n  have S2_sub : S2 x (\u03c8 - \u03a8 R) = S2 x \u03c8 - S2 x (\u03a8 R) := by\n    simp [S2, S1_sub_1] ; rw [integral_sub] ; ring\n    \u00b7 exact \u03c8.integrable_fourier (by positivity) |>.restrict\n    \u00b7 exact (\u03a8 R : W21).integrable_fourier (by positivity) |>.restrict\n\n  have S_sub : S x (\u03c8 - \u03a8 R) = S x \u03c8 - S x (\u03a8 R) := by simp [S, S1_sub, S2_sub] ; ring\n  simpa [S_sub, \u03a8] using norm_add_le _ _ |>.trans_lt (_root_.add_lt_add key3 key)\n\nlemma limiting_cor_schwartz (\u03c8 : \ud835\udce2(\u211d, \u2102)) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) :=\n  limiting_cor_W21 \u03c8 hf hcheby hG hG'\n\n/-%%\n\\begin{proof}\n\\uses{limiting-cor, smooth-ury}\\leanok\nFor any $R>1$, one can use a smooth cutoff function (provided by Lemma \\ref{smooth-ury} to write $\\psi = \\psi_{\\leq R} + \\psi_{>R}$, where $\\psi_{\\leq R}$ is $C^2$ (in fact smooth) and compactly supported (on $[-R,R]$), and $\\psi_{>R}$ obeys bounds of the form\n$$ |\\psi_{>R}(t)|, |\\psi''_{>R}(t)| \\ll R^{-1} / (1 + |t|^2) $$\nwhere the implied constants depend on $\\psi$.  By Lemma \\ref{decay} we then have\n$$ \\hat \\psi_{>R}(u) \\ll R^{-1} / (1+|u|^2).$$\nUsing this and \\eqref{cheby} one can show that\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{>R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ), A \\int_{-\\infty}^\\infty \\hat \\psi_{>R} (\\frac{u}{2\\pi})\\ du \\ll R^{-1} $$\n(with implied constants also depending on $A$), while from Lemma \\ref{limiting-cor} one has\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{\\leq R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi_{\\leq R} (\\frac{u}{2\\pi})\\ du + o(1).$$\nCombining the two estimates and letting $R$ be large, we obtain the claim.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Bijectivity of Fourier transform]\\label{bij}\\lean{fourier_surjection_on_schwartz}\\leanok  The Fourier transform is a bijection on the Schwartz class.\n\\end{lemma}\n%%-/\n\n-- just the surjectivity is stated here, as this is all that is needed for the current application, but perhaps one should state and prove bijectivity instead\n\nlemma fourier_surjection_on_schwartz (f : \ud835\udce2(\u211d, \u2102)) : \u2203 g : \ud835\udce2(\u211d, \u2102), \ud835\udcd5 g = f := by\n  use FS (FS (FS f)) ; ext x ; nth_rewrite 2 [\u2190 FS4 f] ; simp\n\n/-%%\n\\begin{proof}\n  \\leanok\n This is a standard result in Fourier analysis.\nIt can be proved here by appealing to Mellin inversion, Theorem \\ref{MellinInversion}.\nIn particular, given $f$ in the Schwartz class, let $F : \\R_+ \\to \\C : x \\mapsto f(\\log x)$ be a function in the ``Mellin space''; then the Mellin transform of $F$ on the imaginary axis $s=it$ is the Fourier transform of $f$.  The Mellin inversion theorem gives Fourier inversion.\n\\end{proof}\n%%-/\n\ndef toSchwartz (f : \u211d \u2192 \u2102) (h1 : ContDiff \u211d \u22a4 f) (h2 : HasCompactSupport f) : \ud835\udce2(\u211d, \u2102) where\n  toFun := f\n  smooth' := h1\n  decay' k n := by\n    have l1 : Continuous (fun x => \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := by\n      have : ContDiff \u211d \u22a4 (iteratedFDeriv \u211d n f) := h1.iteratedFDeriv_right le_top\n      exact Continuous.mul (by continuity) this.continuous.norm\n    have l2 : HasCompactSupport (fun x \u21a6 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := (h2.iteratedFDeriv _).norm.mul_left\n    simpa using l1.bounded_above_of_compact_support l2\n\n@[simp] lemma toSchwartz_apply (f : \u211d \u2192 \u2102) {h1 h2 x} : SchwartzMap.mk f h1 h2 x = f x := rfl\n\nlemma comp_exp_support0 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in \ud835\udcdd 0, \u03a8 x = 0 :=\n  not_mem_tsupport_iff_eventuallyEq.mp (fun h => lt_irrefl 0 <| mem_Ioi.mp (hplus h))\n\nlemma comp_exp_support1 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in atBot, \u03a8 (exp x) = 0 :=\n  Real.tendsto_exp_atBot <| comp_exp_support0 hplus\n\nlemma comp_exp_support2 {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) :\n    \u2200\u1da0 (x : \u211d) in atTop, (\u03a8 \u2218 rexp) x = 0 := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop] at hsupp\n  exact Real.tendsto_exp_atTop hsupp.2\n\ntheorem comp_exp_support {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    HasCompactSupport (\u03a8 \u2218 rexp) := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop]\n  exact \u27e8comp_exp_support1 hplus, comp_exp_support2 hsupp\u27e9\n\nlemma wiener_ikehara_smooth_aux (l0 : Continuous \u03a8) (hsupp : HasCompactSupport \u03a8)\n    (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) (x : \u211d) (hx : 0 < x) :\n    \u222b (u : \u211d) in Ioi (-Real.log x), \u2191(rexp u) * \u03a8 (rexp u) = \u222b (y : \u211d) in Ioi (1 / x), \u03a8 y := by\n\n  have l1 : ContinuousOn rexp (Ici (-Real.log x)) := by fun_prop\n  have l2 : Tendsto rexp atTop atTop := Real.tendsto_exp_atTop\n  have l3 t (_ : t \u2208 Ioi (-log x)) : HasDerivWithinAt rexp (rexp t) (Ioi t) t :=\n    (Real.hasDerivAt_exp t).hasDerivWithinAt\n  have l4 : ContinuousOn \u03a8 (rexp '' Ioi (-Real.log x)) := by fun_prop\n  have l5 : IntegrableOn \u03a8 (rexp '' Ici (-Real.log x)) volume :=\n    (l0.integrable_of_hasCompactSupport hsupp).integrableOn\n  have l6 : IntegrableOn (fun x \u21a6 rexp x \u2022 (\u03a8 \u2218 rexp) x) (Ici (-Real.log x)) volume := by\n    refine (Continuous.integrable_of_hasCompactSupport (by continuity) ?_).integrableOn\n    change HasCompactSupport (rexp \u2022 (\u03a8 \u2218 rexp))\n    exact (comp_exp_support hsupp hplus).smul_left\n  have := MeasureTheory.integral_comp_smul_deriv_Ioi l1 l2 l3 l4 l5 l6\n  simpa [Real.exp_neg, Real.exp_log hx] using this\n\ntheorem wiener_ikehara_smooth_sub (h1 : Integrable \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    Tendsto (fun x \u21a6 (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n\n  obtain \u27e8\u03b5, h\u03b5, hh\u27e9 := Metric.eventually_nhds_iff.mp <| comp_exp_support0 hplus\n  apply tendsto_nhds_of_eventually_eq ; filter_upwards [eventually_gt_atTop \u03b5\u207b\u00b9] with x hx\u03b5\n\n  have l1 : Integrable (indicator (Ioi x\u207b\u00b9) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n  have l2 : Integrable (indicator (Ioi 0) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n\n  simp_rw [\u2190 MeasureTheory.integral_indicator measurableSet_Ioi, \u2190 mul_sub, \u2190 integral_sub l1 l2]\n  simp ; right ; apply MeasureTheory.integral_eq_zero_of_ae ; apply eventually_of_forall ; intro t ; simp\n\n  have h\u03b5' : 0 < \u03b5\u207b\u00b9 := by positivity\n  have hx : 0 < x := by linarith\n  have hx' : 0 < x\u207b\u00b9 := by positivity\n  have h\u03b5x : x\u207b\u00b9 < \u03b5 := by apply (inv_lt h\u03b5 hx).mp hx\u03b5\n\n  have l3 : Ioi 0 = Ioc 0 x\u207b\u00b9 \u222a Ioi x\u207b\u00b9 := by\n    ext t ; simp ; constructor <;> intro h\n    \u00b7 simp [h, le_or_lt]\n    \u00b7 cases h <;> linarith\n  have l4 : Disjoint (Ioc 0 x\u207b\u00b9) (Ioi x\u207b\u00b9) := by simp\n  have l5 := Set.indicator_union_of_disjoint l4 \u03a8\n  rw [l3, l5] ; ring_nf\n  by_cases ht : t \u2208 Ioc 0 x\u207b\u00b9 <;> simp [ht]\n  apply hh ; simp at ht \u22a2\n  have : |t| \u2264 x\u207b\u00b9 := by rw [abs_le] ; constructor <;> linarith\n  linarith\n\n/-%%\n\\begin{corollary}[Smoothed Wiener-Ikehara]\\label{WienerIkeharaSmooth}\\lean{wiener_ikehara_smooth}\\leanok\n  If $\\Psi: (0,\\infty) \\to \\C$ is smooth and compactly supported away from the origin, then, then\n$$ \\sum_{n=1}^\\infty f(n) \\Psi( \\frac{n}{x} ) = A x \\int_0^\\infty \\Psi(y)\\ dy + o(x)$$\nas $u \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma wiener_ikehara_smooth (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x - A * \u222b y in Set.Ioi 0, \u03a8 y) atTop (nhds 0) := by\n\n  let h (x : \u211d) : \u2102 := rexp (2 * \u03c0 * x) * \u03a8 (exp (2 * \u03c0 * x))\n  have h1 : ContDiff \u211d \u22a4 h := by\n    have : ContDiff \u211d \u22a4 (fun x : \u211d => (rexp (2 * \u03c0 * x))) := (contDiff_const.mul contDiff_id).exp\n    exact (contDiff_ofReal.comp this).mul (hsmooth.comp this)\n  have h2 : HasCompactSupport h := by\n    have : 2 * \u03c0 \u2260 0 := by simp [pi_ne_zero]\n    simpa using (comp_exp_support hsupp hplus).comp_smul this |>.mul_left\n  obtain \u27e8g, hg\u27e9 := fourier_surjection_on_schwartz (toSchwartz h h1 h2)\n\n  have why (x : \u211d) : 2 * \u03c0 * x / (2 * \u03c0) = x := by field_simp ; ring\n  have l1 {y} (hy : 0 < y) : y * \u03a8 y = \ud835\udcd5 g (1 / (2 * \u03c0) * Real.log y) := by\n    field_simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [Real.exp_log hy]\n\n  have key := limiting_cor_schwartz g hf hcheby hG hG'\n\n  have l2 : \u2200\u1da0 x in atTop, \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 (\u21d1g) (1 / (2 * \u03c0) * Real.log (\u2191n / x)) =\n      \u2211' (n : \u2115), f n * \u03a8 (\u2191n / x) / x := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr ; ext n\n    by_cases hn : n = 0 ; simp [hn, (comp_exp_support0 hplus).self_of_nhds]\n    rw [\u2190 l1 (by positivity)]\n    have : (n : \u2102) \u2260 0 := by simpa using hn\n    have : (x : \u2102) \u2260 0 := by simpa using hx.ne.symm\n    field_simp ; ring\n\n  have l3 : \u2200\u1da0 x in atTop, \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 (\u21d1g) (u / (2 * \u03c0)) =\n      \u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr 1 ; simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [MeasureTheory.integral_Ici_eq_integral_Ioi]\n    exact wiener_ikehara_smooth_aux hsmooth.continuous hsupp hplus x hx\n\n  have l4 : Tendsto (fun x => (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n    exact wiener_ikehara_smooth_sub (hsmooth.continuous.integrable_of_hasCompactSupport hsupp) hplus\n\n  simpa [tsum_div_const] using (key.congr' <| EventuallyEq.sub l2 l3) |>.add l4\n\n/-%%\n\\begin{proof}\n\\uses{bij,schwarz-id}\\leanok\n By Lemma \\ref{bij}, we can write\n$$ y \\Psi(y) = \\hat \\psi( \\frac{1}{2\\pi} \\log y )$$\nfor all $y>0$ and some Schwartz function $\\psi$.  Making this substitution, the claim is then equivalent after standard manipulations to\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\nand the claim follows from Lemma \\ref{schwarz-id}.\n\\end{proof}\n%%-/\n\nlemma wiener_ikehara_smooth' (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) :=\n  tendsto_sub_nhds_zero_iff.mp <| wiener_ikehara_smooth hf hcheby hG hG' hsmooth hsupp hplus\n\nlocal instance {E : Type*} : Coe (E \u2192 \u211d) (E \u2192 \u2102) := \u27e8fun f n => f n\u27e9\n\n@[norm_cast]\ntheorem set_integral_ofReal {f : \u211d \u2192 \u211d} {s : Set \u211d} : \u222b x in s, (f x : \u2102) = \u222b x in s, f x :=\n  integral_ofReal\n\nlemma wiener_ikehara_smooth_real {f : \u2115 \u2192 \u211d} {\u03a8 : \u211d \u2192 \u211d} (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) := by\n\n  let \u03a8' := ofReal' \u2218 \u03a8\n  have l1 : ContDiff \u211d \u22a4 \u03a8' := contDiff_ofReal.comp hsmooth\n  have l2 : HasCompactSupport \u03a8' := hsupp.comp_left rfl\n  have l3 : closure (Function.support \u03a8') \u2286 Ioi 0 := by rwa [Function.support_comp_eq] ; simp\n  have key := (continuous_re.tendsto _).comp (@wiener_ikehara_smooth' A \u03a8 G f hf hcheby hG hG' l1 l2 l3)\n  simp at key ; norm_cast at key\n\nlemma interval_approx_inf (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      \u03c8 \u2264 indicator (Ico a b) 1 \u2227 b - a - \u03b5 \u2264 \u222b y in Ioi 0, \u03c8 y := by\n\n  have l1 : Iio ((b - a) / 3) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < (b - a) / 3)\n  have l2 : a < a + \u03b5 / 2 := by linarith\n  have l3 : b - \u03b5 / 2 < b := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 simp [h5, hab.ne, Icc_subset_Ioi_iff hab.le, ha]\n  \u00b7 exact h4.trans <| indicator_le_indicator_of_subset Ioo_subset_Ico_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a - \u03b5 := by linarith\n    have l5 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2229 Ioi 0 = Icc (a + \u03b5 / 2) (b - \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Icc (a + \u03b5 / 2) (b - \u03b5 / 2)) 1 y = b - a - \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; apply set_integral_mono ?_ l8 h3\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Icc]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma interval_approx_sup (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      indicator (Ico a b) 1 \u2264 \u03c8 \u2227 \u222b y in Ioi 0, \u03c8 y \u2264 b - a + \u03b5 := by\n\n  have l1 : Iio (a / 2) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < a / 2)\n  have l2 : a - \u03b5 / 2 < a := by linarith\n  have l3 : b < b + \u03b5 / 2 := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 have l4 : a - \u03b5 / 2 < b + \u03b5 / 2 := by linarith\n    have l5 : \u03b5 / 2 < a := by linarith\n    simp [h5, l4.ne, Icc_subset_Ioi_iff l4.le, l5]\n  \u00b7 apply le_trans ?_ h3\n    apply indicator_le_indicator_of_subset Ico_subset_Icc_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a + \u03b5 := by linarith\n    have l5 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2229 Ioi 0 = Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Ioo (a - \u03b5 / 2) (b + \u03b5 / 2)) 1 y = b - a + \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; refine set_integral_mono l8 ?_ h4\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Ioo]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma WI_summable {f : \u2115 \u2192 \u211d} {g : \u211d \u2192 \u211d} (hg : HasCompactSupport g) (hx : 0 < x) :\n    Summable (fun n => f n * g (n / x)) := by\n  obtain \u27e8M, hM\u27e9 := hg.bddAbove.mono subset_closure\n  apply summable_of_finite_support\n  simp ; apply Finite.inter_of_right ; rw [finite_iff_bddAbove]\n  exact \u27e8Nat.ceil (M * x), fun i hi => by simpa using Nat.ceil_mono ((div_le_iff hx).mp (hM hi))\u27e9\n\nlemma WI_sum_le {f : \u2115 \u2192 \u211d} {g\u2081 g\u2082 : \u211d \u2192 \u211d} (hf : 0 \u2264 f) (hg : g\u2081 \u2264 g\u2082) (hx : 0 < x)\n    (hg\u2081 : HasCompactSupport g\u2081) (hg\u2082 : HasCompactSupport g\u2082) :\n    (\u2211' n, f n * g\u2081 (n / x)) / x \u2264 (\u2211' n, f n * g\u2082 (n / x)) / x := by\n  apply div_le_div_of_nonneg_right ?_ hx.le\n  exact tsum_le_tsum (fun n => mul_le_mul_of_nonneg_left (hg _) (hf _)) (WI_summable hg\u2081 hx) (WI_summable hg\u2082 hx)\n\nlemma WI_sum_Iab_le {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) {C : \u211d} (hcheby : chebyWith C f) (hb : 0 < b) (hxb : 2 / b < x) :\n    (\u2211' n, f n * indicator (Ico a b) 1 (n / x)) / x \u2264 C * 2 * b := by\n  have hb' : 0 < 2 / b := by positivity\n  have hx : 0 < x := by linarith\n  have hxb' : 2 < x * b := (div_lt_iff hb).mp hxb\n  have l1 (i : \u2115) (hi : i \u2209 Finset.range \u2308b * x\u2309\u208a) : f i * indicator (Ico a b) 1 (i / x) = 0 := by\n    simp at hi \u22a2 ; right ; rintro - ; rw [le_div_iff hx] ; linarith\n  have l2 (i : \u2115) (_ : i \u2208 Finset.range \u2308b * x\u2309\u208a) : f i * indicator (Ico a b) 1 (i / x) \u2264 |f i| := by\n    rw [abs_eq_self.mpr (hpos _)]\n    convert_to _ \u2264 f i * 1 ; ring\n    apply mul_le_mul_of_nonneg_left ?_ (hpos _)\n    by_cases hi : (i / x) \u2208 (Ico a b) <;> simp [hi]\n  rw [tsum_eq_sum l1, div_le_iff hx, mul_assoc, mul_assoc]\n  apply Finset.sum_le_sum l2 |>.trans\n  have := hcheby \u2308b * x\u2309\u208a ; simp at this ; apply this.trans\n  have : 0 \u2264 C := by have := hcheby 1 ; simp [cumsum] at this ; exact (abs_nonneg _).trans this\n  refine mul_le_mul_of_nonneg_left ?_ this\n  apply (Nat.ceil_lt_add_one (by positivity)).le.trans\n  linarith\n\nlemma WI_sum_Iab_le' {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) {C : \u211d} (hcheby : chebyWith C f) (hb : 0 < b) :\n    \u2200\u1da0 x : \u211d in atTop, (\u2211' n, f n * indicator (Ico a b) 1 (n / x)) / x \u2264 C * 2 * b := by\n  filter_upwards [eventually_gt_atTop (2 / b)] with x hx using WI_sum_Iab_le hpos hcheby hb hx\n\nlemma le_of_eventually_nhdsWithin {a b : \u211d} (h : \u2200\u1da0 c in \ud835\udcdd[>] b, a \u2264 c) : a \u2264 b := by\n  apply le_of_forall_lt' ; intro d hd\n  have key : \u2200\u1da0 c in \ud835\udcdd[>] b, c < d := by\n    apply eventually_of_mem (U := Iio d) ?_ (fun x hx => hx)\n    rw [mem_nhdsWithin]\n    refine \u27e8Iio d, isOpen_Iio, hd, inter_subset_left _ _\u27e9\n  obtain \u27e8x, h1, h2\u27e9 := (h.and key).exists\n  linarith\n\nlemma ge_of_eventually_nhdsWithin {a b : \u211d} (h : \u2200\u1da0 c in \ud835\udcdd[<] b, c \u2264 a) : b \u2264 a := by\n  apply le_of_forall_lt ; intro d hd\n  have key : \u2200\u1da0 c in \ud835\udcdd[<] b, c > d := by\n    apply eventually_of_mem (U := Ioi d) ?_ (fun x hx => hx)\n    rw [mem_nhdsWithin]\n    refine \u27e8Ioi d, isOpen_Ioi, hd, inter_subset_left _ _\u27e9\n  obtain \u27e8x, h1, h2\u27e9 := (h.and key).exists\n  linarith\n\nlemma WI_tendsto_aux (a b : \u211d) {A : \u211d} (hA : 0 < A) :\n    Tendsto (fun c => c / A - (b - a)) (\ud835\udcdd[>] (A * (b - a))) (\ud835\udcdd[>] 0) := by\n  rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n  intro \u03b5 h\u03b5\n  refine \u27e8A * \u03b5, by positivity, ?_\u27e9\n  intro x hx1 hx2\n  constructor\n  \u00b7 simpa [lt_div_iff' hA]\n  \u00b7 simp [Real.dist_eq] at hx2 \u22a2\n    have : |x / A - (b - a)| = |x - A * (b - a)| / A := by\n      rw [\u2190 abs_eq_self.mpr hA.le, \u2190 abs_div, abs_eq_self.mpr hA.le] ; congr ; field_simp\n    rwa [this, div_lt_iff' hA]\n\nlemma WI_tendsto_aux' (a b : \u211d) {A : \u211d} (hA : 0 < A) :\n    Tendsto (fun c => (b - a) - c / A) (\ud835\udcdd[<] (A * (b - a))) (\ud835\udcdd[>] 0) := by\n  rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n  intro \u03b5 h\u03b5\n  refine \u27e8A * \u03b5, by positivity, ?_\u27e9\n  intro x hx1 hx2\n  constructor\n  \u00b7 simpa [div_lt_iff' hA]\n  \u00b7 simp [Real.dist_eq] at hx2 \u22a2\n    have : |(b - a) - x / A| = |A * (b - a) - x| / A := by\n      rw [\u2190 abs_eq_self.mpr hA.le, \u2190 abs_div, abs_eq_self.mpr hA.le] ; congr ; field_simp ; ring\n    rwa [this, div_lt_iff' hA, \u2190 neg_sub, abs_neg]\n\ntheorem residue_nonneg {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f)\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm (fun n \u21a6 \u2191(f n)) \u03c3')) (hcheby : cheby fun n \u21a6 \u2191(f n))\n    (hG : ContinuousOn G {s | 1 \u2264 s.re}) (hG' : EqOn G (fun s \u21a6 LSeries (fun n \u21a6 \u2191(f n)) s - \u2191A / (s - 1)) {s | 1 < s.re}) : 0 \u2264 A := by\n  let S (g : \u211d \u2192 \u211d) (x : \u211d) := (\u2211' n, f n * g (n / x)) / x\n  have hSnonneg {g : \u211d \u2192 \u211d} (hg : 0 \u2264 g) : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S g x := by\n    filter_upwards [eventually_ge_atTop 0] with x hx\n    exact div_nonneg (tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))) hx\n  obtain \u27e8\u03b5, \u03c8, h1, h2, h3, h4, -\u27e9 := (interval_approx_sup zero_lt_one one_lt_two).exists\n  have key := @wiener_ikehara_smooth_real A G f \u03c8 hf hcheby hG hG' h1 h2 h3\n  have l2 : 0 \u2264 \u03c8 := by apply le_trans _ h4 ; apply indicator_nonneg ; simp\n  have l1 : \u2200\u1da0 x in atTop, 0 \u2264 S \u03c8 x := hSnonneg l2\n  have l3 : 0 \u2264 A * \u222b (y : \u211d) in Ioi 0, \u03c8 y := ge_of_tendsto key l1\n  have l4 : 0 < \u222b (y : \u211d) in Ioi 0, \u03c8 y := by\n    have r1 : 0 \u2264\u1d50[Measure.restrict volume (Ioi 0)] \u03c8 := eventually_of_forall l2\n    have r2 : IntegrableOn (fun y \u21a6 \u03c8 y) (Ioi 0) volume :=\n      (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    have r3 : Ico 1 2 \u2286 Function.support \u03c8 := by intro x hx ; have := h4 x ; simp [hx] at this \u22a2 ; linarith\n    have r4 : Ico 1 2 \u2286 Function.support \u03c8 \u2229 Ioi 0 := by\n      simp [r3] ; apply Ico_subset_Icc_self.trans ; rw [Icc_subset_Ioi_iff] <;> linarith\n    have r5 : 1 \u2264 volume ((Function.support fun y \u21a6 \u03c8 y) \u2229 Ioi 0) := by convert volume.mono r4 ; norm_num\n    simpa [set_integral_pos_iff_support_of_nonneg_ae r1 r2] using zero_lt_one.trans_le r5\n  have := div_nonneg l3 l4.le ; field_simp at this ; exact this\n\n/-%%\nNow we add the hypothesis that $f(n) \\geq 0$ for all $n$.\n\n\\begin{proposition}[Wiener-Ikehara in an interval]\n\\label{WienerIkeharaInterval}\\lean{WienerIkeharaInterval}\\leanok\n  For any closed interval $I \\subset (0,+\\infty)$, we have\n  $$ \\sum_{n=1}^\\infty f(n) 1_I( \\frac{n}{x} ) = A x |I|  + o(x).$$\n\\end{proposition}\n%%-/\n\nlemma WienerIkeharaInterval {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a \u2264 b) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * (indicator (Ico a b) 1 (n / x))) / x) atTop (nhds (A * (b - a))) := by\n\n  -- Take care of the trivial case `a = b`\n  by_cases hab : a = b ; simp [hab] ; replace hb : a < b := lt_of_le_of_ne hb hab ; clear hab\n\n  -- Notation to make the proof more readable\n  let S (g : \u211d \u2192 \u211d) (x : \u211d) :=  (\u2211' n, f n * g (n / x)) / x\n  have hSnonneg {g : \u211d \u2192 \u211d} (hg : 0 \u2264 g) : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S g x := by\n    filter_upwards [eventually_ge_atTop 0] with x hx\n    refine div_nonneg ?_ hx\n    refine tsum_nonneg (fun i => mul_nonneg (hpos _) (hg _))\n  have hA : 0 \u2264 A := residue_nonneg hpos hf hcheby hG hG'\n\n  -- A few facts about the indicator function of `Icc a b`\n  let Iab : \u211d \u2192 \u211d := indicator (Ico a b) 1\n  change Tendsto (S Iab) atTop (\ud835\udcdd (A * (b - a)))\n  have hIab : HasCompactSupport Iab := by simpa [Iab, HasCompactSupport, tsupport, hb.ne] using isCompact_Icc\n  have Iab_nonneg : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 S Iab x := hSnonneg (indicator_nonneg (by simp))\n  have Iab2 : IsBoundedUnder (\u00b7 \u2264 \u00b7) atTop (S Iab) := by\n    obtain \u27e8C, hC\u27e9 := hcheby ; exact \u27e8C * 2 * b, WI_sum_Iab_le' hpos hC (by linarith)\u27e9\n  have Iab3 : IsBoundedUnder (\u00b7 \u2265 \u00b7) atTop (S Iab) := \u27e80, Iab_nonneg\u27e9\n  have Iab0 : IsCoboundedUnder (\u00b7 \u2265 \u00b7) atTop (S Iab) := Iab2.isCoboundedUnder_ge\n  have Iab1 : IsCoboundedUnder (\u00b7 \u2264 \u00b7) atTop (S Iab) := Iab3.isCoboundedUnder_le\n\n  -- Bound from above by a smooth function\n  have sup_le : limsup (S Iab) atTop \u2264 A * (b - a) := by\n    have l_sup : \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, limsup (S Iab) atTop \u2264 A * (b - a + \u03b5) := by\n      filter_upwards [interval_approx_sup ha hb] with \u03b5 \u27e8\u03c8, h1, h2, h3, h4, h6\u27e9\n      have l1 : Tendsto (S \u03c8) atTop _ := wiener_ikehara_smooth_real hf hcheby hG hG' h1 h2 h3\n      have l6 : S Iab \u2264\u1da0[atTop] S \u03c8 := by\n        filter_upwards [eventually_gt_atTop 0] with x hx using WI_sum_le hpos h4 hx hIab h2\n      have l5 : IsBoundedUnder (\u00b7 \u2264 \u00b7) atTop (S \u03c8) := l1.isBoundedUnder_le\n      have l3 : limsup (S Iab) atTop \u2264 limsup (S \u03c8) atTop := limsup_le_limsup l6 Iab1 l5\n      apply l3.trans ; rw [l1.limsup_eq] ; gcongr\n    cases' (eq_or_ne A 0) with h h ; simpa [h] using l_sup\n    apply le_of_eventually_nhdsWithin\n    have key : 0 < A := lt_of_le_of_ne hA h.symm\n    filter_upwards [WI_tendsto_aux a b key l_sup] with x hx\n    simp at hx ; convert hx ; field_simp ; ring\n\n  -- Bound from below by a smooth function\n  have le_inf : A * (b - a) \u2264 liminf (S Iab) atTop := by\n    have l_inf : \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, A * (b - a - \u03b5) \u2264 liminf (S Iab) atTop := by\n      filter_upwards [interval_approx_inf ha hb] with \u03b5 \u27e8\u03c8, h1, h2, h3, h5, h6\u27e9\n      have l1 : Tendsto (S \u03c8) atTop _ := wiener_ikehara_smooth_real hf hcheby hG hG' h1 h2 h3\n      have l2 : S \u03c8 \u2264\u1da0[atTop] S Iab := by\n        filter_upwards [eventually_gt_atTop 0] with x hx using WI_sum_le hpos h5 hx h2 hIab\n      have l4 : IsBoundedUnder (\u00b7 \u2265 \u00b7) atTop (S \u03c8) := l1.isBoundedUnder_ge\n      have l3 : liminf (S \u03c8) atTop \u2264 liminf (S Iab) atTop := liminf_le_liminf l2 l4 Iab0\n      apply le_trans ?_ l3 ; rw [l1.liminf_eq] ; gcongr\n    cases' (eq_or_ne A 0) with h h ; simpa [h] using l_inf\n    apply ge_of_eventually_nhdsWithin\n    have key : 0 < A := lt_of_le_of_ne hA h.symm\n    filter_upwards [WI_tendsto_aux' a b key l_inf] with x hx\n    simp at hx ; convert hx ; field_simp ; ring\n\n  -- Combine the two bounds\n  have : liminf (S Iab) atTop \u2264 limsup (S Iab) atTop := liminf_le_limsup Iab2 Iab3\n  refine tendsto_of_liminf_eq_limsup ?_ ?_ Iab2 Iab3 <;> linarith\n\n/-%%\n\\begin{proof}\n\\uses{smooth-ury, WienerIkeharaSmooth} \\leanok\n  Use Lemma \\ref{smooth-ury} to bound $1_I$ above and below by smooth compactly supported functions whose integral is close to the measure of $|I|$, and use the non-negativity of $f$.\n\\end{proof}\n%%-/\n\nlemma le_floor_mul_iff (hb : 0 \u2264 b) (hx : 0 < x) : n \u2264 \u230ab * x\u230b\u208a \u2194 n / x \u2264 b := by\n  rw [div_le_iff hx, Nat.le_floor_iff] ; positivity\n\nlemma lt_ceil_mul_iff (hx : 0 < x) : n < \u2308b * x\u2309\u208a \u2194 n / x < b := by\n  rw [div_lt_iff hx, Nat.lt_ceil]\n\nlemma ceil_mul_le_iff (hx : 0 < x) : \u2308a * x\u2309\u208a \u2264 n \u2194 a \u2264 n / x := by\n  rw [le_div_iff hx, Nat.ceil_le]\n\nlemma mem_Icc_iff_div (hb : 0 \u2264 b) (hx : 0 < x) : n \u2208 Finset.Icc \u2308a * x\u2309\u208a \u230ab * x\u230b\u208a \u2194 n / x \u2208 Icc a b := by\n  rw [Finset.mem_Icc, mem_Icc, ceil_mul_le_iff hx, le_floor_mul_iff hb hx]\n\nlemma mem_Ico_iff_div (hx : 0 < x) : n \u2208 Finset.Ico \u2308a * x\u2309\u208a \u2308b * x\u2309\u208a \u2194 n / x \u2208 Ico a b := by\n  rw [Finset.mem_Ico, mem_Ico, ceil_mul_le_iff hx, lt_ceil_mul_iff hx]\n\nlemma tsum_indicator {f : \u2115 \u2192 \u211d} (hx : 0 < x) :\n    \u2211' n, f n * (indicator (Ico a b) 1 (n / x)) = \u2211 n in Finset.Ico \u2308a * x\u2309\u208a \u2308b * x\u2309\u208a, f n := by\n  have l1 : \u2200 n \u2209 Finset.Ico \u2308a * x\u2309\u208a \u2308b * x\u2309\u208a, f n * indicator (Ico a b) 1 (\u2191n / x) = 0 := by\n    simp [mem_Ico_iff_div hx] ; tauto\n  rw [tsum_eq_sum l1] ; apply Finset.sum_congr rfl ; simp only [mem_Ico_iff_div hx] ; intro n hn ; simp [hn]\n\nlemma WienerIkeharaInterval_discrete {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a \u2264 b) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211 n in Finset.Ico \u2308a * x\u2309\u208a \u2308b * x\u2309\u208a, f n) / x) atTop (nhds (A * (b - a))) := by\n  apply (WienerIkeharaInterval hpos hf hcheby hG hG' ha hb).congr'\n  filter_upwards [eventually_gt_atTop 0] with x hx\n  rw [tsum_indicator hx]\n\nlemma WienerIkeharaInterval_discrete' {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) (ha : 0 < a) (hb : a \u2264 b) :\n    Tendsto (fun N : \u2115 \u21a6 (\u2211 n in Finset.Ico \u2308a * N\u2309\u208a \u2308b * N\u2309\u208a, f n) / N) atTop (nhds (A * (b - a))) :=\n  WienerIkeharaInterval_discrete hpos hf hcheby hG hG' ha hb |>.comp tendsto_nat_cast_atTop_atTop\n\n-- TODO with `Ico`\n\n/-%%\n\\begin{corollary}[Wiener-Ikehara theorem]\\label{WienerIkehara}\\lean{WienerIkeharaTheorem'}\\leanok\n  We have\n$$ \\sum_{n\\leq x} f(n) = A x |I|  + o(x).$$\n\\end{corollary}\n%%-/\n\n/-- A version of the *Wiener-Ikehara Tauberian Theorem*: If `f` is a nonnegative arithmetic\nfunction whose L-series has a simple pole at `s = 1` with residue `A` and otherwise extends\ncontinuously to the closed half-plane `re s \u2265 1`, then `\u2211 n < N, f n` is asymptotic to `A*N`. -/\n\nlemma tendsto_mul_ceil_div :\n    Tendsto (fun (p : \u211d \u00d7 \u2115) => \u2308p.1 * p.2\u2309\u208a / (p.2 : \u211d)) (\ud835\udcdd[>] 0 \u00d7\u02e2 atTop) (\ud835\udcdd 0) := by\n  rw [Metric.tendsto_nhds] ; intro \u03b4 h\u03b4\n  have l1 : \u2200\u1da0 \u03b5 : \u211d in \ud835\udcdd[>] 0, \u03b5 \u2208 Ioo 0 (\u03b4 / 2) := inter_mem_nhdsWithin _ (Iio_mem_nhds (by positivity))\n  have l2 : \u2200\u1da0 N : \u2115 in atTop, 1 \u2264 \u03b4 / 2 * N := by\n    apply Tendsto.eventually_ge_atTop\n    exact tendsto_nat_cast_atTop_atTop.const_mul_atTop (by positivity)\n  filter_upwards [l1.prod_mk l2] with (\u03b5, N) \u27e8\u27e8h\u03b5, h1\u27e9, h2\u27e9 ; dsimp only at *\n  have l3 : 0 < (N : \u211d) := by\n    simp ; rw [Nat.pos_iff_ne_zero] ; rintro rfl ; simp at h2 ; linarith\n  have l5 : 0 \u2264 \u03b5 * \u2191N := by positivity\n  have l6 : \u03b5 * N \u2264 \u03b4 / 2 * N := mul_le_mul h1.le le_rfl (by positivity) (by positivity)\n  simp [div_lt_iff l3]\n  convert (Nat.ceil_lt_add_one l5).trans_le (add_le_add l6 h2) using 1 ; ring\n\nnoncomputable def S (f : \u2115 \u2192 \ud835\udd5c) (\u03b5 : \u211d) (N : \u2115) : \ud835\udd5c := (\u2211 n in Finset.Ico \u2308\u03b5 * N\u2309\u208a N, f n) / N\n\nlemma S_sub_S {f : \u2115 \u2192 \ud835\udd5c} {\u03b5 : \u211d} {N : \u2115} (h\u03b5 : \u03b5 \u2264 1) : S f 0 N - S f \u03b5 N = cumsum f \u2308\u03b5 * N\u2309\u208a / N := by\n  have r1 : Finset.range N = Finset.range \u2308\u03b5 * N\u2309\u208a \u222a Finset.Ico \u2308\u03b5 * N\u2309\u208a N := by\n    rw [Finset.range_eq_Ico] ; symm ; apply Finset.Ico_union_Ico_eq_Ico (by simp)\n    simp ; convert_to \u03b5 * \u2191N \u2264 1 * \u2191N ; ring ; gcongr\n  have r2 : Disjoint (Finset.range \u2308\u03b5 * N\u2309\u208a) (Finset.Ico \u2308\u03b5 * N\u2309\u208a N) := by\n    rw [Finset.range_eq_Ico] ; apply Finset.Ico_disjoint_Ico_consecutive\n  simp [S, r1, Finset.sum_union r2, cumsum, add_div, abs_div]\n\nlemma tendsto_S_S_zero {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f) (hcheby : cheby f) :\n    TendstoUniformlyOnFilter (S f) (S f 0) (\ud835\udcdd[>] 0) atTop := by\n  rw [Metric.tendstoUniformlyOnFilter_iff] ; intro \u03b4 h\u03b4\n  obtain \u27e8C, hC\u27e9 := hcheby\n  have l1 : \u2200\u1da0 (p : \u211d \u00d7 \u2115) in \ud835\udcdd[>] 0 \u00d7\u02e2 atTop, C * \u2308p.1 * p.2\u2309\u208a / p.2 < \u03b4 := by\n    have r1 := tendsto_mul_ceil_div.const_mul C\n    simp [mul_div_assoc'] at r1 ; exact r1 (Iio_mem_nhds h\u03b4)\n  have : Ioc 0 1 \u2208 \ud835\udcdd[>] (0 : \u211d) := inter_mem_nhdsWithin _ (Iic_mem_nhds zero_lt_one)\n  filter_upwards [l1, Eventually.prod_inl this _] with (\u03b5, N) h1 h2\n  have l2 : |cumsum f \u2308\u03b5 * \u2191N\u2309\u208a / \u2191N| \u2264 C * \u2308\u03b5 * N\u2309\u208a / N := by\n    have r1 := hC \u2308\u03b5 * N\u2309\u208a\n    have r2 : 0 \u2264 cumsum f \u2308\u03b5 * N\u2309\u208a := by apply cumsum_nonneg hpos\n    simp [abs_div, abs_eq_self.mpr r2, abs_eq_self.mpr (hpos _)] at r1 \u22a2\n    apply div_le_div_of_nonneg_right r1 (by positivity)\n  simpa [\u2190 S_sub_S h2.2] using l2.trans_lt h1\n\n", "theoremStatement": "theorem WienerIkeharaTheorem' {f : \u2115 \u2192 \u211d} (hpos : 0 \u2264 f)\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun N => cumsum f N / N) atTop (\ud835\udcdd A) ", "theoremName": "WienerIkeharaTheorem'", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "a5f2c75771039fea1eee34479fe21e690876e6f7", "date": "2024-01-23"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 2099, "tokenPositionInFile": 106514, "theoremPositionInFile": 166}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 176, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", 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"PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n\n  convert_to Tendsto (S f 0) atTop (\ud835\udcdd A) ; \u00b7 ext N ; simp [S, cumsum]\n  apply (tendsto_S_S_zero hpos hcheby).tendsto_of_eventually_tendsto\n  \u00b7 have L0 : Ioc 0 1 \u2208 \ud835\udcdd[>] (0 : \u211d) := inter_mem_nhdsWithin _ (Iic_mem_nhds zero_lt_one)\n    apply eventually_of_mem L0 ; intro \u03b5 h\u03b5\n    simpa using WienerIkeharaInterval_discrete' hpos hf hcheby hG hG' h\u03b5.1 h\u03b5.2\n  \u00b7 have : Tendsto (fun \u03b5 : \u211d => \u03b5) (\ud835\udcdd[>] 0) (\ud835\udcdd 0) := nhdsWithin_le_nhds\n    simpa using (this.const_sub 1).const_mul A", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 479}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.MeasureTheory.Integral.IntervalIntegral\nimport Mathlib.Analysis.Calculus.Deriv.Basic\nimport Mathlib.NumberTheory.ZetaFunction\nimport Mathlib.Algebra.Group.Basic\nimport EulerProducts.PNT\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.MellinCalculus\nimport Mathlib.MeasureTheory.Function.Floor\nimport Mathlib.Analysis.Complex.CauchyIntegral\nimport Mathlib.NumberTheory.Harmonic.Bounds\n\n-- only importing the following for the MeasurableDiv\u2082 \u211d instance.\n-- should remove eventually\nimport PrimeNumberTheoremAnd.PerronFormula\n\n-- set_option quotPrecheck false\nopen BigOperators Complex Topology Filter Interval Set\n\nlemma div_cpow_eq_cpow_neg (a x s : \u2102) : a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, cpow_neg]\n\nlemma one_div_cpow_eq_cpow_neg (x s : \u2102) : 1 / x ^ s = x ^ (-s) := by\n  convert div_cpow_eq_cpow_neg 1 x s using 1; simp\n\nlemma div_rpow_eq_rpow_neg (a x s : \u211d) (hx : 0 \u2264 x): a / x ^ s = a * x ^ (-s) := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx]\n\nlemma div_rpow_neg_eq_rpow_div {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ (-s) / y ^ (-s) = (y / x) ^ s := by\n  rw [div_eq_mul_inv, Real.rpow_neg hx, Real.rpow_neg hy, Real.div_rpow hy hx]; field_simp\n\n", "theoremStatement": "lemma div_rpow_eq_rpow_div_neg {x y s : \u211d} (hx : 0 \u2264 x) (hy : 0 \u2264 y) :\n    x ^ s / y ^ s = (y / x) ^ (-s) ", "theoremName": "div_rpow_eq_rpow_div_neg", "fileCreated": {"commit": "cefb8d3426335f4b0dc0ca6d2acdecdad256648b", "date": "2024-02-28"}, "theoremCreated": {"commit": "fa5d9115712e016e4dfb55d5de4242d509838224", "date": "2024-04-24"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/ZetaBounds.lean", "module": "PrimeNumberTheoremAnd.ZetaBounds", "jsonFile": "PrimeNumberTheoremAnd.ZetaBounds.jsonl", "positionMetadata": {"lineInFile": 33, "tokenPositionInFile": 1268, 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"Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula", "PrimeNumberTheoremAnd.MellinCalculus", "Mathlib.MeasureTheory.Function.Floor", "Mathlib.NumberTheory.Harmonic.Defs", "Mathlib.NumberTheory.Harmonic.Bounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  convert div_rpow_neg_eq_rpow_div (s := -s) hx hy using 1; simp only [neg_neg]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 85}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\nlemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u := by ext n ; simp [nabla, nnabla]\n\n@[simp] lemma nabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nabla (fun n => c * u n) = c \u2022 nabla u := by\n  ext n ; simp [nabla, mul_sub]\n\n@[simp] lemma nnabla_mul [Ring E] {u : \u03b1 \u2192 E} {c : E} : nnabla (fun n => c * u n) = c \u2022 nnabla u := by\n  ext n ; simp [nnabla, mul_sub]\n\nlemma nnabla_cast (u : \u211d \u2192 E) [Sub E] : nnabla u \u2218 ((\u2191) : \u2115 \u2192 \u211d) = nnabla (u \u2218 (\u2191)) := by\n  ext n ; simp [nnabla]\n\nend nabla\n\nlemma Finset.sum_shift_front {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = u 0 + cumsum (shift u) n := by\n  simp_rw [add_comm n, cumsum, sum_range_add, sum_range_one, add_comm 1] ; rfl\n\nlemma Finset.sum_shift_front' {E : Type*} [Ring E] {u : \u2115 \u2192 E} :\n    shift (cumsum u) = (fun _ => u 0) + cumsum (shift u) := by\n  ext n ; apply Finset.sum_shift_front\n\nlemma Finset.sum_shift_back {E : Type*} [Ring E] {u : \u2115 \u2192 E} {n : \u2115} :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.range_succ, add_comm]\n\nlemma Finset.sum_shift_back' {E : Type*} [Ring E] {u : \u2115 \u2192 E} : shift (cumsum u) = cumsum u + u := by\n  ext n ; apply Finset.sum_shift_back\n\nlemma summation_by_parts {E : Type*} [Ring E] {a A b : \u2115 \u2192 E} (ha : a = nabla A) {n : \u2115} :\n    cumsum (a * b) (n + 1) = A (n + 1) * b n - A 0 * b 0 - cumsum (shift A * fun i => (b (i + 1) - b i)) n := by\n  have l1 : \u2211 x in Finset.range (n + 1), A (x + 1) * b x = \u2211 x in Finset.range n, A (x + 1) * b x + A (n + 1) * b n :=\n    Finset.sum_shift_back\n  have l2 : \u2211 x in Finset.range (n + 1), A x * b x = A 0 * b 0 + \u2211 x in Finset.range n, A (x + 1) * b (x + 1) :=\n    Finset.sum_shift_front\n  simp [cumsum, shift, ha, nabla, sub_mul, mul_sub, l1, l2] ; abel\n\nlemma summation_by_parts' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} {n : \u2115} :\n    cumsum (a * b) (n + 1) = cumsum a (n + 1) * b n - cumsum (shift (cumsum a) * nabla b) n := by\n  simpa using summation_by_parts (a := a) (b := b) (A := cumsum a) (by simp [Finset.sum_shift_back])\n\nlemma summation_by_parts'' {E : Type*} [Ring E] {a b : \u2115 \u2192 E} :\n    shift (cumsum (a * b)) = shift (cumsum a) * b - cumsum (shift (cumsum a) * nabla b) := by\n  ext n ; apply summation_by_parts'\n\nlemma summable_iff_bounded {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  have l1 : (cumsum u =O[atTop] 1) \u2194 _ := isBigO_one_nat_atTop_iff\n  have l2 n : \u2016cumsum u n\u2016 = cumsum u n := by simpa using cumsum_nonneg hu n\n  simp only [BoundedAtFilter, l1, l2]\n  constructor <;> intro \u27e8C, h1\u27e9\n  \u00b7 exact \u27e8C, fun n => sum_le_hasSum _ (fun i _ => hu i) h1\u27e9\n  \u00b7 exact summable_of_sum_range_le hu h1\n\nlemma Filter.EventuallyEq.summable {u v : \u2115 \u2192 \u211d} (h : u =\u1da0[atTop] v) (hu : Summable v) : Summable u :=\n  summable_of_isBigO_nat hu h.isBigO\n\nlemma summable_congr_ae {u v : \u2115 \u2192 \u211d} (huv : u =\u1da0[atTop] v) : Summable u \u2194 Summable v := by\n  constructor <;> intro h <;> simp [huv.summable, huv.symm.summable, h]\n\nlemma BoundedAtFilter.add_const {u : \u2115 \u2192 \u211d} {c : \u211d} :\n    BoundedAtFilter atTop (fun n => u n + c) \u2194 BoundedAtFilter atTop u := by\n  have : u = fun n => (u n + c) + (-c) := by ext n ; ring\n  simp [BoundedAtFilter] ; constructor <;> intro h ; rw [this]\n  all_goals { exact h.add (const_boundedAtFilter _ _) }\n\nlemma BoundedAtFilter.comp_add {u : \u2115 \u2192 \u211d} {N : \u2115} :\n    BoundedAtFilter atTop (fun n => u (n + N)) \u2194 BoundedAtFilter atTop u := by\n  simp [BoundedAtFilter, isBigO_iff] ; constructor <;> intro \u27e8C, n\u2080, h\u27e9 <;> use C\n  \u00b7 refine \u27e8n\u2080 + N, fun n hn => ?_\u27e9\n    obtain \u27e8k, hk\u27e9 := Nat.exists_eq_add_of_le' (m := N) (by linarith) ; subst n\n    exact h _ <| Nat.add_le_add_iff_right.mp hn\n  \u00b7 exact \u27e8n\u2080, fun n hn => h _ (by linarith)\u27e9\n\nlemma summable_iff_bounded' {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, 0 \u2264 u n) :\n    Summable u \u2194 BoundedAtFilter atTop (cumsum u) := by\n  obtain \u27e8N, hu\u27e9 := eventually_atTop.mp hu\n  have e2 : cumsum (fun i \u21a6 u (i + N)) = fun n => cumsum u (n + N) - cumsum u N := by\n    ext n ; simp_rw [cumsum, add_comm _ N, Finset.sum_range_add] ; ring\n  rw [\u2190 summable_nat_add_iff N, summable_iff_bounded (fun n => hu _ <| Nat.le_add_left N n), e2]\n  simp_rw [sub_eq_add_neg, BoundedAtFilter.add_const, BoundedAtFilter.comp_add]\n\nlemma bounded_of_shift {u : \u2115 \u2192 \u211d} (h : BoundedAtFilter atTop (shift u)) : BoundedAtFilter atTop u := by\n  simp only [BoundedAtFilter, isBigO_iff, eventually_atTop] at h \u22a2\n  obtain \u27e8C, N, hC\u27e9 := h\n  refine \u27e8C, N + 1, fun n hn => ?_\u27e9\n  simp only [shift] at hC\n  have r1 : n - 1 \u2265 N := Nat.le_sub_one_of_lt hn\n  have r2 : n - 1 + 1 = n := Nat.sub_add_cancel <| NeZero.one_le.trans hn.le\n  simpa [r2] using hC (n - 1) r1\n\nlemma dirichlet_test' {a b : \u2115 \u2192 \u211d} (ha : 0 \u2264 a) (hb : 0 \u2264 b)\n    (hAb : BoundedAtFilter atTop (shift (cumsum a) * b)) (hbb : \u2200\u1da0 n in atTop, b (n + 1) \u2264 b n)\n    (h : Summable (shift (cumsum a) * nnabla b)) : Summable (a * b) := by\n  have l1 : \u2200\u1da0 n in atTop, 0 \u2264 (shift (cumsum a) * nnabla b) n := by\n    filter_upwards [hbb] with n hb\n    exact mul_nonneg (by simpa [shift] using Finset.sum_nonneg' ha) (sub_nonneg.mpr hb)\n  rw [summable_iff_bounded (mul_nonneg ha hb)]\n  rw [summable_iff_bounded' l1] at h\n  apply bounded_of_shift\n  simpa only [summation_by_parts'', sub_eq_add_neg, neg_cumsum, \u2190 mul_neg, neg_nabla] using hAb.add h\n\nlemma exists_antitone_of_eventually {u : \u2115 \u2192 \u211d} (hu : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n) :\n    \u2203 v : \u2115 \u2192 \u211d, range v \u2286 range u \u2227 Antitone v \u2227 v =\u1da0[atTop] u := by\n  obtain \u27e8N, hN\u27e9 := eventually_atTop.mp hu\n  let v (n : \u2115) := u (if n < N then N else n)\n  refine \u27e8v, ?_, ?_, ?_\u27e9\n  \u00b7 exact fun x \u27e8n, hn\u27e9 => \u27e8if n < N then N else n, hn\u27e9\n  \u00b7 refine antitone_nat_of_succ_le (fun n => ?_)\n    by_cases h : n < N\n    \u00b7 by_cases h' : n + 1 < N <;> simp [v, h, h']\n      have : n + 1 = N := by linarith\n      simp [this]\n    \u00b7 have : \u00ac(n + 1 < N) := by linarith\n      simp [v, h, this] ; apply hN ; linarith\n  \u00b7 have : \u2200\u1da0 n in atTop, \u00ac(n < N) := by simpa using \u27e8N, fun b hb => by linarith\u27e9\n    filter_upwards [this] with n hn ; simp [v, hn]\n\nlemma summable_inv_mul_log_sq : Summable (fun n : \u2115 => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n  let u (n : \u2115) := (n * (Real.log n) ^ 2)\u207b\u00b9\n  have l7 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 Real.log n := tendsto_atTop.mp (tendsto_log_atTop.comp tendsto_nat_cast_atTop_atTop) 1\n  have l8 : \u2200\u1da0 n : \u2115 in atTop, 1 \u2264 n := eventually_ge_atTop 1\n  have l9 : \u2200\u1da0 n in atTop, u (n + 1) \u2264 u n := by filter_upwards [l7, l8] with n l2 l8 ; dsimp [u] ; gcongr <;> simp\n  obtain \u27e8v, l1, l2, l3\u27e9 := exists_antitone_of_eventually l9\n  rw [summable_congr_ae l3.symm]\n  have l4 (n : \u2115) : 0 \u2264 v n := by obtain \u27e8k, hk\u27e9 := l1 \u27e8n, rfl\u27e9 ; rw [\u2190 hk] ; positivity\n  apply (summable_condensed_iff_of_nonneg l4 (fun _ _ _ a \u21a6 l2 a)).mp\n  suffices this : \u2200\u1da0 k : \u2115 in atTop, 2 ^ k * v (2 ^ k) = ((k : \u211d) ^ 2)\u207b\u00b9 * ((Real.log 2) ^ 2)\u207b\u00b9 by\n    exact (summable_congr_ae this).mpr <| (Real.summable_nat_pow_inv.mpr one_lt_two).mul_right _\n  have l5 : \u2200\u1da0 k in atTop, v (2 ^ k) = u (2 ^ k) := l3.comp_tendsto <| Nat.tendsto_pow_atTop_atTop_of_one_lt Nat.le.refl\n  filter_upwards [l5, l8] with k l5 l8 ; field_simp [u, l5] ; ring\n\nlemma tendsto_mul_add_atTop {a : \u211d} (ha : 0 < a) (b : \u211d) : Tendsto (fun x => a * x + b) atTop atTop :=\n  tendsto_atTop_add_const_right  _ b (tendsto_id.const_mul_atTop ha)\n\nlemma isLittleO_const_of_tendsto_atTop {\u03b1 : Type*} [Preorder \u03b1] (a : \u211d) {f : \u03b1 \u2192 \u211d} (hf : Tendsto f atTop atTop) :\n    (fun _ => a) =o[atTop] f := by\n  simp [tendsto_norm_atTop_atTop.comp hf]\n\nlemma isBigO_pow_pow_of_le {m n : \u2115} (h : m \u2264 n) : (fun x : \u211d => x ^ m) =O[atTop] (fun x : \u211d => x ^ n) := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_ge_atTop 1] with x l1\n  simpa [abs_eq_self.mpr (zero_le_one.trans l1)] using pow_le_pow_right l1 h\n\nlemma isLittleO_mul_add_sq (a b : \u211d) : (fun x => a * x + b) =o[atTop] (fun x => x ^ 2) := by\n  apply IsLittleO.add\n  \u00b7 apply IsLittleO.const_mul_left ; simpa using isLittleO_pow_pow_atTop_of_lt (\ud835\udd5c := \u211d) one_lt_two\n  \u00b7 apply isLittleO_const_of_tendsto_atTop _ <| tendsto_pow_atTop (by linarith)\n\nlemma log_mul_add_isBigO_log {a : \u211d} (ha : 0 < a) (b : \u211d) : (fun x => Real.log (a * x + b)) =O[atTop] Real.log := by\n  apply IsBigO.of_bound (2 : \u2115)\n  have l2 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log x := tendsto_atTop.mp tendsto_log_atTop 0\n  have l3 : \u2200\u1da0 x : \u211d in atTop, 0 \u2264 log (a * x + b) :=\n    tendsto_atTop.mp (tendsto_log_atTop.comp (tendsto_mul_add_atTop ha b)) 0\n  have l5 : \u2200\u1da0 x : \u211d in atTop, 1 \u2264 a * x + b := tendsto_atTop.mp (tendsto_mul_add_atTop ha b) 1\n  have l1 : \u2200\u1da0 x : \u211d in atTop, a * x + b \u2264 x ^ 2 := by\n    filter_upwards [(isLittleO_mul_add_sq a b).eventuallyLE, l5] with x r2 l5\n    simpa [abs_eq_self.mpr (zero_le_one.trans l5)] using r2\n  filter_upwards [l1, l2, l3, l5] with x l1 l2 l3 l5\n  simpa [abs_eq_self.mpr l2, abs_eq_self.mpr l3, Real.log_pow] using Real.log_le_log (by linarith) l1\n\nlemma isBigO_log_mul_add {a : \u211d} (ha : 0 < a) (b : \u211d) : Real.log =O[atTop] (fun x => Real.log (a * x + b)) := by\n  convert (log_mul_add_isBigO_log (b := -b / a) (inv_pos.mpr ha)).comp_tendsto (tendsto_mul_add_atTop (b := b) ha) using 1\n  ext x ; field_simp [ha.ne.symm] ; rw [mul_div_assoc, mul_div_cancel\u2080] ; linarith\n\nlemma log_isbigo_log_div {d : \u211d} (hb : 0 < d) : (fun n \u21a6 Real.log n) =O[atTop] (fun n \u21a6 Real.log (n / d)) := by\n  convert isBigO_log_mul_add (inv_pos.mpr hb) 0 using 1 ; field_simp\n\nlemma Asymptotics.IsBigO.add_isLittleO_right {f g : \u211d \u2192 \u211d} (h : g =o[atTop] f) : f =O[atTop] (f + g) := by\n  rw [isLittleO_iff] at h ; specialize h (c := 2\u207b\u00b9) (by norm_num)\n  rw [isBigO_iff''] ; refine \u27e82\u207b\u00b9, by norm_num, ?_\u27e9 ; filter_upwards [h] with x h ; simp at h \u22a2\n  calc _ = |f x| - 2\u207b\u00b9 * |f x| := by ring\n       _ \u2264 |f x| - |g x| := by linarith\n       _ \u2264 |(|f x| - |g x|)| := le_abs_self _\n       _ \u2264 _ := by rw [\u2190 sub_neg_eq_add, \u2190 abs_neg (g x)] ; exact abs_abs_sub_abs_le (f x) (-g x)\n\nlemma Asymptotics.IsBigO.sq {\u03b1 : Type*} [Preorder \u03b1] {f g : \u03b1 \u2192 \u211d} (h : f =O[atTop] g) :\n    (fun n \u21a6 f n ^ 2) =O[atTop] (fun n => g n ^ 2) := by\n  simpa [pow_two] using h.mul h\n\nlemma log_sq_isbigo_mul {a b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log x ^ 2) =O[atTop] (fun x \u21a6 a + Real.log (x / b) ^ 2) := by\n  apply (log_isbigo_log_div hb).sq.trans ; simp_rw [add_comm a]\n  refine IsBigO.add_isLittleO_right <| isLittleO_const_of_tendsto_atTop _ ?_\n  exact (tendsto_pow_atTop (two_ne_zero)).comp <| tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\ntheorem log_add_div_isBigO_log (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    (fun x \u21a6 Real.log ((x + a) / b)) =O[atTop] fun x \u21a6 Real.log x := by\n  convert log_mul_add_isBigO_log (inv_pos.mpr hb) (a / b) using 3 ; ring\n\nlemma log_add_one_sub_log_le {x : \u211d} (hx : 0 < x) : nabla Real.log x \u2264 x\u207b\u00b9 := by\n  have l1 : ContinuousOn Real.log (Icc x (x + 1)) := by\n    apply continuousOn_log.mono ; intro t \u27e8h1, _\u27e9 ; simp ; linarith\n  have l2 t (ht : t \u2208 Ioo x (x + 1)) : HasDerivAt Real.log t\u207b\u00b9 t := Real.hasDerivAt_log (by linarith [ht.1])\n  obtain \u27e8t, \u27e8ht1, _\u27e9, htx\u27e9 := exists_hasDerivAt_eq_slope Real.log (\u00b7\u207b\u00b9) (by linarith) l1 l2\n  simp at htx ; rw [nabla, \u2190 htx, inv_le_inv (by linarith) hx] ; linarith\n\nlemma nabla_log_main : nabla Real.log =O[atTop] fun x \u21a6 1 / x := by\n  apply IsBigO.of_bound 1\n  filter_upwards [eventually_gt_atTop 0] with x l1\n  have l2 : log x \u2264 log (x + 1) := log_le_log l1 (by linarith)\n  simpa [nabla, abs_eq_self.mpr l1.le, abs_eq_self.mpr (sub_nonneg.mpr l2)] using log_add_one_sub_log_le l1\n\nlemma nabla_log {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => Real.log (x / b)) =O[atTop] (fun x => 1 / x) := by\n  refine EventuallyEq.trans_isBigO ?_ nabla_log_main\n  filter_upwards [eventually_gt_atTop 0] with x l2\n  rw [nabla, log_div (by linarith) (by linarith), log_div l2.ne.symm (by linarith), nabla] ; ring\n\nlemma nnabla_mul_log_sq (a : \u211d) {b : \u211d} (hb : 0 < b) :\n    nabla (fun x => x * (a + Real.log (x / b) ^ 2)) =O[atTop] (fun x => Real.log x ^ 2) := by\n\n  have l1 : nabla (fun n => n * (a + Real.log (n / b) ^ 2)) = fun n =>\n      a + Real.log ((n + 1) / b) ^ 2 + (n * (Real.log ((n + 1) / b) ^ 2 - Real.log (n / b) ^ 2)) := by\n    ext n ; simp [nabla] ; ring\n  have l2 := (isLittleO_const_of_tendsto_atTop a ((tendsto_pow_atTop two_ne_zero).comp tendsto_log_atTop)).isBigO\n  have l3 := (log_add_div_isBigO_log 1 hb).sq\n  have l4 : (fun x => Real.log ((x + 1) / b) + Real.log (x / b)) =O[atTop] Real.log := by\n    simpa using (log_add_div_isBigO_log _ hb).add (log_add_div_isBigO_log 0 hb)\n  have e2 : (fun x : \u211d => x * (Real.log x * (1 / x))) =\u1da0[atTop] Real.log := by\n    filter_upwards [eventually_ge_atTop 1] with x hx ; field_simp ; ring\n  have l5 : (fun n \u21a6 n * (Real.log n * (1 / n))) =O[atTop] (fun n \u21a6 (Real.log n) ^ 2) :=\n    e2.trans_isBigO (by simpa using (isLittleO_mul_add_sq 1 0).isBigO.comp_tendsto Real.tendsto_log_atTop)\n\n  simp_rw [l1, _root_.sq_sub_sq]\n  exact ((l2.add l3).add (isBigO_refl (\u00b7) atTop |>.mul (l4.mul (nabla_log hb)) |>.trans l5))\n\nlemma nnabla_bound_aux1 (a : \u211d) {b : \u211d} (hb : 0 < b) : Tendsto (fun x => x * (a + Real.log (x / b) ^ 2)) atTop atTop :=\n  tendsto_id.atTop_mul_atTop <| tendsto_atTop_add_const_left _ _ <| (tendsto_pow_atTop two_ne_zero).comp <|\n    tendsto_log_atTop.comp <| tendsto_id.atTop_div_const hb\n\nlemma nnabla_bound_aux2 (a : \u211d) {b : \u211d} (hb : 0 < b) : \u2200\u1da0 x in atTop, 0 < x * (a + Real.log (x / b) ^ 2) :=\n  (nnabla_bound_aux1 a hb).eventually (eventually_gt_atTop 0)\n\nlemma nnabla_bound_aux {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n \u21a6 1 / (n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2))) =O[atTop]\n    (fun n \u21a6 1 / (Real.log n ^ 2 * n ^ 2)) := by\n\n  let d n : \u211d := n * ((2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2)\n  change (fun x_1 \u21a6 nnabla (fun n \u21a6 1 / d n) x_1) =O[atTop] _\n\n  have l2 : \u2200\u1da0 n in atTop, 0 < d n := (nnabla_bound_aux2 ((2 * \u03c0) ^ 2) hx)\n  have l3 : \u2200\u1da0 n in atTop, 0 < d (n + 1) :=\n    (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually l2\n  have l1 : \u2200\u1da0 n : \u211d in atTop, nnabla (fun n \u21a6 1 / d n) n = (d (n + 1) - d n) * (d n)\u207b\u00b9 * (d (n + 1))\u207b\u00b9 := by\n    filter_upwards [l2, l3] with n l2 l3\n    rw [nnabla, one_div, one_div, inv_sub_inv l2.ne.symm l3.ne.symm, div_eq_mul_inv, mul_inv, mul_assoc]\n\n  have l4 : (fun n => (d n)\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    apply IsBigO.inv_rev\n    \u00b7 refine (isBigO_refl _ _).mul <| (log_sq_isbigo_mul (by linarith))\n    \u00b7 apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n      have e1 : n \u2260 0 := by linarith\n      have e2 : n \u2260 1 := by linarith\n      have e3 : n \u2260 -1 := by linarith\n      simp [e1, e2, e3]\n\n  have l5 : (fun n => (d (n + 1))\u207b\u00b9) =O[atTop] (fun n => (n * (Real.log n) ^ 2)\u207b\u00b9) := by\n    refine IsBigO.trans ?_ l4\n    rw [isBigO_iff] ; use 1\n    have e1 : \u2200\u1da0 n in atTop, 0 < d n := by\n      apply eventually_of_mem (Ici_mem_atTop 1) ; intro n (hn : 1 \u2264 n)\n      have r1 : 0 < n := by linarith\n      have r2 : 0 < (2 * \u03c0) ^ 2 := by apply sq_pos_of_ne_zero ; norm_num [pi_ne_zero]\n      have r3 : 0 \u2264 Real.log (\u2191n / x) ^ 2 := sq_nonneg _\n      apply mul_pos r1 (by linarith)\n    have e2 : \u2200\u1da0 n in atTop, 0 < d (n + 1) := (tendsto_atTop_add_const_right atTop (1 : \u211d) tendsto_id).eventually e1\n    have e3 : \u2200\u1da0 n in atTop, d n \u2264 d (n + 1) := by\n      have : \u2200\u1da0 n in atTop, x \u2264 n := by simpa using eventually_ge_atTop x\n      filter_upwards [this] with n hn\n      have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n      have e3 : n \u2264 n + 1 := by linarith\n      have e4 : 0 \u2264 n + 1 := by linarith\n      dsimp [d]\n      gcongr\n      exact Real.log_nonneg e2\n    filter_upwards [e1, e2, e3] with n e1 e2 e3\n    simp_rw [one_mul, Real.norm_eq_abs, abs_inv, abs_eq_self.mpr e1.le, abs_eq_self.mpr e2.le, inv_le_inv e2 e1]\n    exact e3\n\n  have l6 : (fun n => d (n + 1) - d n) =O[atTop] (fun n => (Real.log n) ^ 2) := by\n    simpa [d, nabla] using (nnabla_mul_log_sq ((2 * \u03c0) ^ 2) (by linarith))\n\n  apply EventuallyEq.trans_isBigO l1\n\n  apply ((l6.mul l4).mul l5).trans_eventuallyEq\n  apply eventually_of_mem (Ici_mem_atTop 2) ; intro n (hn : 2 \u2264 n)\n\n  have : Real.log n \u2260 0 := by\n    have e1 : n \u2260 0 := by linarith\n    have e2 : n \u2260 1 := by linarith\n    have e3 : n \u2260 -1 := by linarith\n    simp [e1, e2, e3]\n  field_simp ; ring\n\nlemma nnabla_bound (C : \u211d) {x : \u211d} (hx : 0 < x) :\n    nnabla (fun n => C / (1 + (Real.log (n / x) / (2 * \u03c0)) ^ 2) / n) =O[atTop]\n    (fun n => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n  field_simp\n  simp [div_eq_mul_inv]\n  apply IsBigO.const_mul_left\n  field_simp\n  exact nnabla_bound_aux hx\n\ndef chebyWith (C : \u211d) (f : \u2115 \u2192 \u2102) : Prop := \u2200 n, cumsum (\u2016f \u00b7\u2016) n \u2264 C * n\n\ndef cheby (f : \u2115 \u2192 \u2102) : Prop := \u2203 C, chebyWith C f\n\nlemma cheby.bigO (h : cheby f) : cumsum (\u2016f \u00b7\u2016) =O[atTop] ((\u2191) : \u2115 \u2192 \u211d) := by\n  have l1 : 0 \u2264 cumsum (\u2016f \u00b7\u2016) := cumsum_nonneg (fun _ => norm_nonneg _)\n  obtain \u27e8C, hC\u27e9 := h\n  apply isBigO_of_le' (c := C) atTop\n  intro n\n  rw [Real.norm_eq_abs, abs_eq_self.mpr (l1 n)]\n  simpa using hC n\n\nlemma limiting_fourier_lim1_aux (hcheby : cheby f) (hx : 0 < x) (C : \u211d) (hC : 0 \u2264 C) :\n    Summable fun n \u21a6 \u2016f n\u2016 / \u2191n * (C / (1 + (1 / (2 * \u03c0) * Real.log (\u2191n / x)) ^ 2)) := by\n\n  let a (n : \u2115) := (C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2) / \u2191n)\n  replace hcheby := hcheby.bigO\n\n  have l1 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n : \u2115 => (\u2191(n + 1) : \u211d)) :=\n    hcheby.comp_tendsto <| tendsto_add_atTop_nat 1\n  have l2 : shift (cumsum (\u2016f \u00b7\u2016)) =O[atTop] (fun n => (n : \u211d)) :=\n    l1.trans (by simpa using (isBigO_refl _ _).add <| isBigO_iff.mpr \u27e81, by simpa using \u27e81, by tauto\u27e9\u27e9)\n  have l5 : BoundedAtFilter atTop (fun n : \u2115 => C / (1 + (Real.log (\u2191n / x) / (2 * \u03c0)) ^ 2)) := by\n    field_simp [BoundedAtFilter]\n    apply isBigO_of_le' (c := C) ; intro n\n    have : 0 \u2264 (2 * \u03c0) ^ 2 + Real.log (n / x) ^ 2 := by positivity\n    simp [abs_eq_self.mpr hC, abs_eq_self.mpr pi_nonneg, abs_eq_self.mpr this]\n    apply div_le_of_nonneg_of_le_mul this hC\n    gcongr\n    apply le_add_of_le_of_nonneg le_rfl (sq_nonneg _)\n  have l3 : a =O[atTop] (fun n => 1 / (n : \u211d)) := by\n    simpa [a] using IsBigO.mul l5 (isBigO_refl (fun n : \u2115 => 1 / (n : \u211d)) _)\n  have l4 : nnabla a =O[atTop] (fun n : \u2115 => (n ^ 2 * (Real.log n) ^ 2)\u207b\u00b9) := by\n    convert (nnabla_bound C hx).natCast ; simp [nnabla, a]\n\n  simp_rw [div_mul_eq_mul_div, mul_div_assoc, one_mul]\n  apply dirichlet_test'\n  \u00b7 intro n ; exact norm_nonneg _\n  \u00b7 intro n ; positivity\n  \u00b7 apply (l2.mul l3).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 1)\n    intro x (hx : 1 \u2264 x)\n    have : x \u2260 0 := by linarith\n    simp [this]\n  \u00b7 have : \u2200\u1da0 n : \u2115 in atTop, x \u2264 n := by simpa using eventually_ge_atTop \u2308x\u2309\u208a\n    filter_upwards [this] with n hn\n    have e1 : 0 < (n : \u211d) := by linarith\n    have e2 : 1 \u2264 n / x := (one_le_div (by linarith)).mpr hn\n    have e3 := Nat.le_succ n\n    gcongr\n    refine div_nonneg (Real.log_nonneg e2) (by norm_num [pi_nonneg])\n  \u00b7 apply summable_of_isBigO_nat summable_inv_mul_log_sq\n    apply (l2.mul l4).trans_eventuallyEq\n    apply eventually_of_mem (Ici_mem_atTop 2)\n    intro x (hx : 2 \u2264 x)\n    have : (x : \u211d) \u2260 0 := by simp ; linarith\n    have : Real.log x \u2260 0 := by\n      have ll : 2 \u2264 (x : \u211d) := by simp [hx]\n      simp only [ne_eq, log_eq_zero]\n      push_neg\n      refine \u27e8this, ?_, ?_\u27e9 <;> linarith\n    field_simp ; ring\n\ntheorem limiting_fourier_lim1 (hcheby : cheby f) (\u03c8 : W21) (hx : 0 < x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x))) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (n / x)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  have : 0 \u2264 C := by simpa using (norm_nonneg _).trans (hC 0)\n  refine tendsto_tsum_of_dominated_convergence (limiting_fourier_lim1_aux hcheby hx C this) (fun n => ?_) ?_\n  \u00b7 apply Tendsto.mul_const\n    by_cases h : n = 0 <;> simp [term, h]\n    refine tendsto_const_nhds.div ?_ (by simp [h])\n    simpa using ((continuous_ofReal.tendsto 1).mono_left nhdsWithin_le_nhds).const_cpow\n  \u00b7 rw [eventually_nhdsWithin_iff]\n    apply eventually_of_forall\n    intro \u03c3' (h\u03c3' : 1 < \u03c3') n\n    rw [norm_mul, \u2190 nterm_eq_norm_term]\n    refine mul_le_mul ?_ (hC _) (norm_nonneg _) (div_nonneg (norm_nonneg _) (Nat.cast_nonneg _))\n    by_cases h : n = 0 <;> simp [h, nterm]\n    have : 1 \u2264 (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr h\n    refine div_le_div (by simp only [apply_nonneg]) le_rfl (by simpa [Nat.pos_iff_ne_zero]) ?_\n    simpa using Real.rpow_le_rpow_of_exponent_le this h\u03c3'.le\n\ntheorem limiting_fourier_lim2_aux (x : \u211d) (C : \u211d) :\n    Integrable (fun t \u21a6 |x| * (C / (1 + (t / (2 * \u03c0)) ^ 2))) (Measure.restrict volume (Ici (-Real.log x))) := by\n  simp_rw [div_eq_mul_inv C]\n  exact (((integrable_inv_one_add_sq.comp_div (by simp [pi_ne_zero])).const_mul _).const_mul _).restrict\n\ntheorem limiting_fourier_lim2 (A : \u211d) (\u03c8 : W21) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' \u21a6 A * \u2191(x ^ (1 - \u03c3')) * \u222b u in Ici (-Real.log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))\n      (\ud835\udcdd[>] 1) (\ud835\udcdd (A * \u222b u in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)))) := by\n\n  obtain \u27e8C, hC\u27e9 := decay_bounds_cor \u03c8\n  apply Tendsto.mul\n  \u00b7 suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 ofReal' (x ^ (1 - \u03c3'))) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.const_mul \u2191A\n    suffices h : Tendsto (fun \u03c3' : \u211d \u21a6 x ^ (1 - \u03c3')) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) from (continuous_ofReal.tendsto 1).comp h\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 \u03c3') (\ud835\udcdd 1) (\ud835\udcdd 1) := fun _ a \u21a6 a\n    have : Tendsto (fun \u03c3' : \u211d \u21a6 1 - \u03c3') (\ud835\udcdd[>] 1) (\ud835\udcdd 0) :=\n      tendsto_nhdsWithin_of_tendsto_nhds (by simpa using this.const_sub 1)\n    simpa using tendsto_const_nhds.rpow this (Or.inl (zero_lt_one.trans_le hx).ne.symm)\n  \u00b7 refine tendsto_integral_filter_of_dominated_convergence _ ?_ ?_ (limiting_fourier_lim2_aux x C) ?_\n    \u00b7 apply eventually_of_forall ; intro \u03c3'\n      apply Continuous.aestronglyMeasurable\n      have := continuous_FourierIntegral \u03c8\n      continuity\n    \u00b7 apply eventually_of_mem (U := Ioo 1 2)\n      \u00b7 apply Ioo_mem_nhdsWithin_Ioi ; simp\n      \u00b7 intro \u03c3' \u27e8h1, h2\u27e9\n        rw [ae_restrict_iff' measurableSet_Ici]\n        apply eventually_of_forall\n        intro t (ht : - Real.log x \u2264 t)\n        rw [norm_mul]\n        refine mul_le_mul ?_ (hC _) (norm_nonneg _) (abs_nonneg _)\n        simp [Complex.abs_exp]\n        have : -Real.log x * (\u03c3' - 1) \u2264 t * (\u03c3' - 1) := mul_le_mul_of_nonneg_right ht (by linarith)\n        have : -(t * (\u03c3' - 1)) \u2264 Real.log x * (\u03c3' - 1) := by simpa using neg_le_neg this\n        have := Real.exp_monotone this\n        apply this.trans\n        have l1 : \u03c3' - 1 \u2264 1 := by linarith\n        have : 0 \u2264 Real.log x := Real.log_nonneg hx\n        have := mul_le_mul_of_nonneg_left l1 this\n        apply (Real.exp_monotone this).trans\n        simp [Real.exp_log (zero_lt_one.trans_le hx), abs_eq_self.mpr (zero_le_one.trans hx)]\n    \u00b7 apply eventually_of_forall\n      intro x\n      suffices h : Tendsto (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) (\ud835\udcdd[>] 1) (\ud835\udcdd 1) by simpa using h.mul_const _\n      apply Tendsto.mono_left ?_ nhdsWithin_le_nhds\n      suffices h : Continuous (fun n \u21a6 ((rexp (-x * (n - 1))) : \u2102)) by simpa using h.tendsto 1\n      continuity\n\ntheorem limiting_fourier_lim3 (hG : ContinuousOn G {s | 1 \u2264 s.re}) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    Tendsto (fun \u03c3' : \u211d \u21a6 \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I)) (\ud835\udcdd[>] 1)\n      (\ud835\udcdd (\u222b t : \u211d, G (1 + t * I) * \u03c8 t * x ^ (t * I))) := by\n\n  by_cases hh : tsupport \u03c8 = \u2205 ; simp [tsupport_eq_empty_iff.mp hh]\n  obtain \u27e8a\u2080, ha\u2080\u27e9 := Set.nonempty_iff_ne_empty.mpr hh\n\n  let S : Set \u2102 := Set.reProdIm (Icc 1 2) (tsupport \u03c8)\n  have l1 : IsCompact S := by\n    refine Metric.isCompact_iff_isClosed_bounded.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact isClosed_Icc.reProdIm (isClosed_tsupport \u03c8)\n    \u00b7 exact (Metric.isBounded_Icc 1 2).reProdIm \u03c8.h2.isBounded\n  have l2 : S \u2286 {s : \u2102 | 1 \u2264 s.re} := fun z hz => (mem_reProdIm.mp hz).1.1\n  have l3 : ContinuousOn (\u2016G \u00b7\u2016) S := (hG.mono l2).norm\n  have l4 : S.Nonempty := \u27e81 + a\u2080 * I, by simp [S, mem_reProdIm, ha\u2080]\u27e9\n  obtain \u27e8z, -, hmax\u27e9 := l1.exists_isMaxOn l4 l3\n  let MG := \u2016G z\u2016\n  let bound (a : \u211d) : \u211d := MG * \u2016\u03c8 a\u2016\n\n  apply tendsto_integral_filter_of_dominated_convergence (bound := bound)\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp)) ; intro u hu\n    apply Continuous.aestronglyMeasurable\n    apply Continuous.mul\n    \u00b7 exact (hG.comp_continuous (by continuity) (by simp [hu.1])).mul \u03c8.h1.continuous\n    \u00b7 apply Continuous.const_cpow (by continuity) ; simp ; linarith\n  \u00b7 apply eventually_of_mem (U := Icc 1 2) (Icc_mem_nhdsWithin_Ioi (by simp))\n    intro u hu\n    apply eventually_of_forall ; intro v\n    by_cases h : v \u2208 tsupport \u03c8\n    \u00b7 have r1 : u + v * I \u2208 S := by simp [S, mem_reProdIm, hu.1, hu.2, h]\n      have r2 := isMaxOn_iff.mp hmax _ r1\n      have r4 : (x : \u2102) \u2260 0 := by simp ; linarith\n      have r5 : arg x = 0 := by simp [arg_eq_zero_iff] ; linarith\n      have r3 : \u2016(x : \u2102) ^ (v * I)\u2016 = 1 := by simp [abs_cpow_of_ne_zero r4, r5]\n      simp_rw [norm_mul, r3, mul_one]\n      exact mul_le_mul_of_nonneg_right r2 (norm_nonneg _)\n    \u00b7 have : v \u2209 Function.support \u03c8 := fun a \u21a6 h (subset_tsupport \u03c8 a)\n      simp at this ; simp [this, bound]\n\n  \u00b7 suffices h : Continuous bound by exact h.integrable_of_hasCompactSupport \u03c8.h2.norm.mul_left\n    have := \u03c8.h1.continuous ; continuity\n  \u00b7 apply eventually_of_forall ; intro t\n    apply Tendsto.mul_const\n    apply Tendsto.mul_const\n    refine (hG (1 + t * I) (by simp)).tendsto.comp <| tendsto_nhdsWithin_iff.mpr \u27e8?_, ?_\u27e9\n    \u00b7 exact ((continuous_ofReal.tendsto _).add tendsto_const_nhds).mono_left nhdsWithin_le_nhds\n    \u00b7 exact eventually_nhdsWithin_of_forall (fun x (hx : 1 < x) => by simp [hx.le])\n\nlemma limiting_fourier (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) :\n    \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n      \u222b (t : \u211d), (G (1 + t * I)) * (\u03c8 t) * x ^ (t * I) := by\n\n  have l1 := limiting_fourier_lim1 hcheby \u03c8 (by linarith)\n  have l2 := limiting_fourier_lim2 A \u03c8 hx\n  have l3 := limiting_fourier_lim3 hG \u03c8 hx\n  apply tendsto_nhds_unique_of_eventuallyEq (l1.sub l2) l3\n  simpa [eventuallyEq_nhdsWithin_iff] using eventually_of_forall (limiting_fourier_aux hG' hf \u03c8 hx)\n\n/-%%\n\\begin{proof}\n\\uses{first-fourier,second-fourier,decay} \\leanok\n By the preceding two lemmas, we know that for any $\\sigma>1$, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A x^{1-\\sigma} \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(\\sigma+it) \\psi(t) x^{it}\\ dt.$$\n  Now take limits as $\\sigma \\to 1$ using dominated convergence together with \\eqref{cheby} and Lemma \\ref{decay} to obtain the result.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{corollary}[Corollary of limiting identity]\\label{limiting-cor}\\lean{limiting_cor}\\leanok  With the hypotheses as above, we have\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\n  as $x \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma limiting_cor_aux {f : \u211d \u2192 \u2102} : Tendsto (fun x : \u211d \u21a6 \u222b t, f t * x ^ (t * I)) atTop (\ud835\udcdd 0) := by\n\n  have l1 : \u2200\u1da0 x : \u211d in atTop, \u2200 t : \u211d, x ^ (t * I) = exp (log x * t * I) := by\n    filter_upwards [eventually_ne_atTop 0, eventually_ge_atTop 0] with x hx hx' t\n    rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr hx), ofReal_log hx'] ; ring_nf\n\n  have l2 : \u2200\u1da0 x : \u211d in atTop, \u222b t, f t * x ^ (t * I) = \u222b t, f t * exp (log x * t * I) := by\n    filter_upwards [l1] with x hx\n    refine integral_congr_ae (eventually_of_forall (fun x => by simp [hx]))\n\n  simp_rw [tendsto_congr' l2]\n  convert_to Tendsto (fun x => \ud835\udcd5 f (-Real.log x / (2 * \u03c0))) atTop (\ud835\udcdd 0)\n  \u00b7 ext ; congr ; ext ; simp [Real.fourierChar, expMapCircle, mul_comm (f _)] ; left ; congr\n    rw [\u2190 neg_mul] ; congr ; norm_cast ; field_simp ; ring\n  refine (zero_at_infty_fourierIntegral f).comp <| Tendsto.mono_right ?_ _root_.atBot_le_cocompact\n  exact (tendsto_neg_atBot_iff.mpr tendsto_log_atTop).atBot_mul_const (inv_pos.mpr two_pi_pos)\n\nlemma limiting_cor (\u03c8 : CS 2 \u2102) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (nhds 0) := by\n\n  apply limiting_cor_aux.congr'\n  filter_upwards [eventually_ge_atTop 1] with x hx using limiting_fourier hcheby hG hG' hf \u03c8 hx |>.symm\n\n/-%%\n\\begin{proof}\n\\uses{limiting} \\leanok\n Immediate from the Riemann-Lebesgue lemma, and also noting that $\\int_{-\\infty}^{-\\log x} \\hat \\psi(\\frac{u}{2\\pi})\\ du = o(1)$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Smooth Urysohn lemma]\\label{smooth-ury}\\lean{smooth_urysohn}\\leanok  If $I$ is a closed interval contained in an open interval $J$, then there exists a smooth function $\\Psi: \\R \\to \\R$ with $1_I \\leq \\Psi \\leq 1_J$.\n\\end{lemma}\n%%-/\n\nlemma smooth_urysohn (a b c d : \u211d) (h1 : a < b) (h3 : c < d) : \u2203 \u03a8 : \u211d \u2192 \u211d,\n    (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227\n      Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227 \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 := by\n\n  obtain \u27e8\u03c8, l1, l2, l3, l4, -\u27e9 := smooth_urysohn_support_Ioo h1 h3\n  refine \u27e8\u03c8, l1, l2, l3, l4\u27e9\n\n/-%%\n\\begin{proof}  \\leanok\nA standard analysis lemma, which can be proven by convolving $1_K$ with a smooth approximation to the identity for some interval $K$ between $I$ and $J$. Note that we have ``SmoothBumpFunction''s on smooth manifolds in Mathlib, so this shouldn't be too hard...\n\\end{proof}\n%%-/\n\nnoncomputable def exists_trunc : trunc := by\n  choose \u03c8 h1 h2 h3 h4 using smooth_urysohn (-2) (-1) (1) (2) (by linarith) (by linarith)\n  exact \u27e8\u27e8\u03c8, h1.of_le le_top, h2\u27e9, h3, h4\u27e9\n\nlemma one_div_sub_one (n : \u2115) : 1 / (\u2191(n - 1) : \u211d) \u2264 2 / n := by\n  match n with\n  | 0 => simp\n  | 1 => simp\n  | n + 2 => { norm_cast ; rw [div_le_div_iff] <;> simp [mul_add] <;> linarith }\n\nlemma quadratic_pos (a b c x : \u211d) (ha : 0 < a) (h\u0394 : discrim a b c < 0) : 0 < a * x ^ 2 + b * x + c := by\n  have l1 : a * x ^ 2 + b * x + c = a * (x + b / (2 * a)) ^ 2 - discrim a b c / (4 * a) := by\n    field_simp [discrim] ; ring\n  have l2 : 0 < - discrim a b c := by linarith\n  rw [l1, sub_eq_add_neg, \u2190 neg_div] ; positivity\n\nnoncomputable def pp (a x : \u211d) : \u211d := a ^ 2 * (x + 1) ^ 2 + (1 - a) * (1 + a)\n\nnoncomputable def pp' (a x : \u211d) : \u211d := a ^ 2 * (2 * (x + 1))\n\nlemma pp_pos {a : \u211d} (ha : a \u2208 Ioo (-1) 1) (x : \u211d) : 0 < pp a x := by\n  simp [pp]\n  have : 0 < 1 - a := by linarith [ha.2]\n  have : 0 < 1 + a := by linarith [ha.1]\n  positivity\n\nlemma pp_deriv (a x : \u211d) : HasDerivAt (pp a) (pp' a x) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.pow 2 |>.const_mul _ |>.add_const _\n\nlemma pp_deriv_eq (a : \u211d) : deriv (pp a) = pp' a := by\n  ext x ; exact pp_deriv a x |>.deriv\n\nlemma pp'_deriv (a x : \u211d) : HasDerivAt (pp' a) (a ^ 2 * 2) x := by\n  simpa using hasDerivAt_id x |>.add_const 1 |>.const_mul 2 |>.const_mul (a ^ 2)\n\nlemma pp'_deriv_eq (a : \u211d) : deriv (pp' a) = fun _ => a ^ 2 * 2 := by\n  ext x ; exact pp'_deriv a x |>.deriv\n\nnoncomputable def hh (a t : \u211d) : \u211d := (t * (1 + (a * log t) ^ 2))\u207b\u00b9\n\nnoncomputable def hh' (a t : \u211d) : \u211d := - pp a (log t) * hh a t ^ 2\n\nlemma hh_nonneg (a : \u211d) {t : \u211d} (ht : 0 \u2264 t) : 0 \u2264 hh a t := by dsimp only [hh] ; positivity\n\nlemma hh_le (a t : \u211d) (ht : 0 \u2264 t) : |hh a t| \u2264 t\u207b\u00b9 := by\n  by_cases h0 : t = 0 ; simp [hh, h0]\n  replace ht : 0 < t := lt_of_le_of_ne ht (by tauto)\n  unfold hh\n  rw [abs_inv, inv_le_inv (by positivity) ht, abs_mul, abs_eq_self.mpr ht.le]\n  convert_to t * 1 \u2264 _ ; simp\n  apply mul_le_mul le_rfl ?_ zero_le_one ht.le\n  rw [abs_eq_self.mpr (by positivity)]\n  simp ; positivity\n\nlemma hh_deriv (a : \u211d) {t : \u211d} (ht : t \u2260 0) : HasDerivAt (hh a) (hh' a t) t := by\n  have e1 : t * (1 + (a * log t) ^ 2) \u2260 0 := mul_ne_zero ht (_root_.ne_of_lt (by positivity)).symm\n  have l5 : HasDerivAt (fun t : \u211d => log t) t\u207b\u00b9 t := Real.hasDerivAt_log ht\n  have l4 : HasDerivAt (fun t : \u211d => a * log t) (a * t\u207b\u00b9) t := l5.const_mul _\n  have l3 : HasDerivAt (fun t : \u211d => (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := by\n    convert l4.pow 2 using 1 ; ring\n  have l2 : HasDerivAt (fun t : \u211d => 1 + (a * log t) ^ 2) (2 * a ^ 2 * t\u207b\u00b9 * log t) t := l3.const_add _\n  have l1 : HasDerivAt (fun t : \u211d => t * (1 + (a * log t) ^ 2))\n      (1 + 2 * a ^ 2 * log t + a ^ 2 * log t ^ 2) t := by\n    convert (hasDerivAt_id t).mul l2 using 1 ; field_simp ; ring\n  convert l1.inv e1 using 1 ; field_simp [hh', hh, pp] ; ring\n\nlemma hh_continuous (a : \u211d) : ContinuousOn (hh a) (Ioi 0) :=\n  fun t (ht : 0 < t) => (hh_deriv a ht.ne.symm).continuousAt.continuousWithinAt\n\nlemma hh'_nonpos {a x : \u211d} (ha : a \u2208 Ioo (-1) 1) : hh' a x \u2264 0 := by\n  have := pp_pos ha (log x)\n  have := hh_nonneg a (sq_nonneg x)\n  simp [hh'] ; positivity\n\nlemma hh_antitone {a : \u211d} (ha : a \u2208 Ioo (-1) 1) : AntitoneOn (hh a) (Ioi 0) := by\n  have l1 x (hx : x \u2208 interior (Ioi 0)) : HasDerivWithinAt (hh a) (hh' a x) (interior (Ioi 0)) x := by\n    have : x \u2260 0 := by contrapose! hx ; simp [hx]\n    exact (hh_deriv a this).hasDerivWithinAt\n  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi _) (hh_continuous _) l1 (fun x _ => hh'_nonpos ha)\n\nnoncomputable def gg (x i : \u211d) : \u211d := 1 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9\n\nlemma gg_of_hh {x : \u211d} (hx : x \u2260 0) (i : \u211d) : gg x i = x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (i / x) := by\n  by_cases hi : i = 0 ; simp [gg, hh, hi]\n  field_simp [gg, hh] ; ring\n\nlemma gg_l1 {x : \u211d} (hx : 0 < x) (n : \u2115) : |gg x n| \u2264 1 / n := by\n  simp [gg_of_hh hx.ne.symm, abs_mul]\n  apply mul_le_mul le_rfl (hh_le _ _ (by positivity)) (by positivity) (by positivity) |>.trans (le_of_eq ?_)\n  simp [abs_inv, abs_eq_self.mpr hx.le] ; field_simp\n\nlemma gg_le_one (i : \u2115) : gg x i \u2264 1 := by\n  by_cases hi : i = 0 <;> simp [gg, hi]\n  have l1 : 1 \u2264 (i : \u211d) := by simp ; omega\n  have l2 : 1 \u2264 1 + (\u03c0\u207b\u00b9 * 2\u207b\u00b9 * Real.log (\u2191i / x)) ^ 2 := by simp ; positivity\n  rw [\u2190 mul_inv] ; apply inv_le_one ; simpa using mul_le_mul l1 l2 zero_le_one (by simp)\n\nlemma one_div_two_pi_mem_Ioo : 1 / (2 * \u03c0) \u2208 Ioo (-1) 1 := by\n  constructor\n  \u00b7 trans 0 ; linarith ; positivity\n  \u00b7 rw [div_lt_iff (by positivity)]\n    convert_to 1 * 1 < 2 * \u03c0 ; simp ; simp\n    apply mul_lt_mul one_lt_two ?_ zero_lt_one zero_le_two\n    trans 2 ; exact one_le_two ; exact two_le_pi\n\nlemma sum_telescopic (a : \u2115 \u2192 \u211d) (n : \u2115) : \u2211 i in Finset.range n, (a (i + 1) - a i) = a n - a 0 := by\n  apply Finset.sum_range_sub\n\nlemma cancel_aux {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264 g (n - 1) * (C * n) + (C * (\u2191(n - 1 - 1) + 1) * g 0\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1) -\n    ((n - 1 - 1) \u2022 (C * g 0) - \u2211 x in Finset.range (n - 1 - 1), C * g (x + 1))) := by\n\n  have l1 (n : \u2115) : (g n - g (n + 1)) * \u2211 i in Finset.range (n + 1), f i \u2264 (g n - g (n + 1)) * (C * (n + 1)) := by\n    apply mul_le_mul le_rfl (by simpa using hf' (n + 1)) (Finset.sum_nonneg' hf) ?_\n    simp ; apply hg' ; simp\n  have l2 (x : \u2115) : C * (\u2191(x + 1) + 1) - C * (\u2191x + 1) = C := by simp ; ring\n  have l3 (n : \u2115) : 0 \u2264 cumsum f n := Finset.sum_nonneg' hf\n\n  convert_to \u2211 i in Finset.range n, (g i) \u2022 (f i) \u2264 _ ; simp [mul_comm]\n  rw [Finset.sum_range_by_parts, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n  simp_rw [\u2190 neg_smul, neg_sub, smul_eq_mul]\n  apply _root_.add_le_add\n  \u00b7 exact mul_le_mul le_rfl (hf' n) (l3 n) (hg _)\n  \u00b7 apply Finset.sum_le_sum (fun n _ => l1 n) |>.trans\n    convert_to \u2211 i in Finset.range (n - 1), (C * (\u2191i + 1)) \u2022 (g i - g (i + 1)) \u2264 _\n    \u00b7 congr ; ext i ; simp ; ring\n    rw [Finset.sum_range_by_parts]\n    simp_rw [Finset.sum_range_sub', l2, smul_sub, smul_eq_mul, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range]\n    apply le_of_eq ; ring_nf\n\nlemma sum_range_succ (a : \u2115 \u2192 \u211d) (n : \u2115) :\n    \u2211 i in Finset.range n, a (i + 1) = (\u2211 i in Finset.range (n + 1), a i) - a 0 := by\n  have := Finset.sum_range_sub a n\n  rw [Finset.sum_sub_distrib, sub_eq_iff_eq_add] at this\n  rw [Finset.sum_range_succ, this] ; ring\n\nlemma cancel_aux' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    \u2211 i in Finset.range n, f i * g i \u2264\n        C * n * g (n - 1)\n      + C * cumsum g (n - 1 - 1 + 1)\n      - C * (\u2191(n - 1 - 1) + 1) * g (n - 1)\n      := by\n  have := cancel_aux hf hg hf' hg' n ; simp [\u2190 Finset.mul_sum, sum_range_succ] at this\n  convert this using 1 ; unfold cumsum ; ring\n\nlemma cancel_main {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) (hn : 2 \u2264 n) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  convert cancel_aux' hf hg hf' hg' n using 1\n  match n with\n  | n + 2 => simp [cumsum_succ] ; ring\n\nlemma cancel_main' {C : \u211d} {f g : \u2115 \u2192 \u211d} (hf : 0 \u2264 f) (hf0 : f 0 = 0) (hg : 0 \u2264 g)\n    (hf' : \u2200 n, cumsum f n \u2264 C * n) (hg' : Antitone g) (n : \u2115) :\n    cumsum (f * g) n \u2264 C * cumsum g n := by\n  match n with\n  | 0 => simp [cumsum]\n  | 1 => specialize hg 0 ; specialize hf' 1 ; simp [cumsum, hf0] at hf' hg \u22a2 ; positivity\n  | n + 2 => convert cancel_aux' hf hg hf' hg' (n + 2) using 1 ; simp [cumsum_succ] ; ring\n\ntheorem sum_le_integral {x\u2080 : \u211d} {f : \u211d \u2192 \u211d} {n : \u2115} (hf : AntitoneOn f (Ioc x\u2080 (x\u2080 + n)))\n    (hfi : IntegrableOn f (Icc x\u2080 (x\u2080 +  n))) :\n    (\u2211 i in Finset.range n, f (x\u2080 + \u2191(i + 1))) \u2264 \u222b x in x\u2080..x\u2080 + n, f x := by\n\n  cases' n with n <;> simp [Nat.succ_eq_add_one] at hf \u22a2\n  have : Finset.range (n + 1) = {0} \u222a Finset.Ico 1 (n + 1) := by\n    ext i ; by_cases hi : i = 0 <;> simp [hi] ; omega\n  simp [this, Finset.sum_union]\n\n  have l4 : IntervalIntegrable f volume x\u2080 (x\u2080 + 1) := by\n    apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  have l5 x (hx : x \u2208 Ioc x\u2080 (x\u2080 + 1)) : (fun x \u21a6 f (x\u2080 + 1)) x \u2264 f x := by\n    rcases hx with \u27e8hx1, hx2\u27e9\n    refine hf \u27e8hx1, by linarith\u27e9 \u27e8by linarith, by linarith\u27e9 hx2\n  have l6 : \u222b x in x\u2080..x\u2080 + 1, f (x\u2080 + 1) = f (x\u2080 + 1) := by simp\n\n  have l1 : f (x\u2080 + 1) \u2264 \u222b x in x\u2080..x\u2080 + 1, f x := by\n    rw [\u2190 l6] ; apply intervalIntegral.integral_mono_ae_restrict (by linarith) (by simp) l4\n    apply eventually_of_mem _ l5\n    have : (Ioc x\u2080 (x\u2080 + 1))\u1d9c \u2229 Icc x\u2080 (x\u2080 + 1) = {x\u2080} := by simp [\u2190 diff_eq_compl_inter]\n    simp [Measure.ae, this]\n\n  have l2 : AntitoneOn (fun x \u21a6 f (x\u2080 + x)) (Icc 1 \u2191(n + 1)) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8_, hv2\u27e9 huv ; push_cast at hv2\n    refine hf \u27e8?_, ?_\u27e9 \u27e8?_, ?_\u27e9 ?_ <;> linarith\n\n  have l3 := @AntitoneOn.sum_le_integral_Ico 1 (n + 1) (fun x => f (x\u2080 + x)) (by simp) (by simpa using l2)\n\n  simp at l3\n  convert _root_.add_le_add l1 l3\n\n  have := @intervalIntegral.integral_comp_mul_add \u211d _ _ 1 (n + 1) 1 f one_ne_zero x\u2080\n  simp [add_comm _ x\u2080] at this ; rw [this]\n  rw [intervalIntegral.integral_add_adjacent_intervals]\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n  \u00b7 apply IntegrableOn.intervalIntegrable\n    simp\n    apply hfi.mono_set\n    apply Icc_subset_Icc ; linarith ; simp\n\nlemma hh_integrable_aux (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    (IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0)) \u2227\n    (\u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0) := by\n\n  simp only [integrableOn_Ici_iff_integrableOn_Ioi, hh]\n\n  let g (x : \u211d) := (a * c / b) * arctan (b * log (x / c))\n  let g\u2080 (x : \u211d) := if x = 0 then ((a * c / b) * (- (\u03c0 / 2))) else g x\n  let g' (x : \u211d) := a * (x / c * (1 + (b * Real.log (x / c)) ^ 2))\u207b\u00b9\n\n  have l3 (x) (hx : 0 < x) : HasDerivAt Real.log x\u207b\u00b9 x := by apply Real.hasDerivAt_log (by linarith)\n  have l4 (x) : HasDerivAt (fun t => t / c) (1 / c) x := (hasDerivAt_id x).div_const c\n  have l2 (x) (hx : 0 < x) : HasDerivAt (fun t => log (t / c)) x\u207b\u00b9 x := by\n    have := @HasDerivAt.comp _ _ _ _ _ _ (fun t => t / c) _ _ _  (l3 (x / c) (by positivity)) (l4 x)\n    convert this using 1 ; field_simp ; ring\n  have l5 (x) (hx : 0 < x) := (l2 x hx).const_mul b\n  have l1 (x) (hx : 0 < x) := (l5 x hx).arctan\n  have l6 (x) (hx : 0 < x) : HasDerivAt g (g' x) x := by\n    convert (l1 x hx).const_mul (a * c / b) using 1\n    field_simp [g'] ; ring\n  have key (x) (hx : 0 < x) : HasDerivAt g\u2080 (g' x) x := by\n    apply (l6 x hx).congr_of_eventuallyEq\n    apply eventually_of_mem <| Ioi_mem_nhds hx\n    intro y (hy : 0 < y)\n    simp [g\u2080, hy.ne.symm]\n\n  have k1 : Tendsto g\u2080 atTop (\ud835\udcdd ((a * c / b) * (\u03c0 / 2))) := by\n    have : g =\u1da0[atTop] g\u2080 := by\n      apply eventually_of_mem (Ioi_mem_atTop 0)\n      intro y (hy : 0 < y)\n      simp [g\u2080, hy.ne.symm]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atTop hb\n    apply tendsto_log_atTop.comp\n    apply Tendsto.atTop_div_const hc\n    apply tendsto_id\n\n  have k2 : Tendsto g\u2080 (\ud835\udcdd[>] 0) (\ud835\udcdd (g\u2080 0)) := by\n    have : g =\u1da0[\ud835\udcdd[>] 0] g\u2080 := by\n      apply eventually_of_mem self_mem_nhdsWithin\n      intro x (hx : 0 < x) ; simp [g\u2080, hx.ne.symm]\n    simp only [g\u2080]\n    apply Tendsto.congr' this\n    apply Tendsto.const_mul\n    apply (tendsto_arctan_atBot.mono_right nhdsWithin_le_nhds).comp\n    apply Tendsto.const_mul_atBot hb\n    apply tendsto_log_nhdsWithin_zero_right.comp\n    rw [Metric.tendsto_nhdsWithin_nhdsWithin]\n    intro \u03b5 h\u03b5\n    refine \u27e8c * \u03b5, by positivity, fun hx1 hx2 => \u27e8?_, ?_\u27e9\u27e9\n    \u00b7 simp at hx1 \u22a2 ; positivity\n    \u00b7 simp [abs_eq_self.mpr hc.le] at hx2 \u22a2 ; rwa [div_lt_iff hc, mul_comm]\n\n  have k3 : ContinuousWithinAt g\u2080 (Ici 0) 0 := by\n    rw [Metric.continuousWithinAt_iff]\n    rw [Metric.tendsto_nhdsWithin_nhds] at k2\n    peel k2 with \u03b5 h\u03b5 \u03b4 h\u03b4 x h\n    intro (hx : 0 \u2264 x)\n    have := le_iff_lt_or_eq.mp hx\n    cases this with\n    | inl hx => exact h hx\n    | inr hx => simp [g\u2080, hx.symm, h\u03b5]\n\n  have k4 : \u2200 x \u2208 Ioi 0, 0 \u2264 g' x := by\n    intro x (hx : 0 < x) ; simp [g'] ; positivity\n\n  constructor\n  \u00b7 convert_to IntegrableOn g' _\n    exact integrableOn_Ioi_deriv_of_nonneg k3 key k4 k1\n  \u00b7 have := integral_Ioi_of_hasDerivAt_of_nonneg k3 key k4 k1\n    simp [g\u2080, g'] at this \u22a2\n    convert this using 1 ; field_simp ; ring\n\nlemma hh_integrable (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    IntegrableOn (fun t \u21a6 a * hh b (t / c)) (Ici 0) :=\n  hh_integrable_aux ha hb hc |>.1\n\nlemma hh_integral (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :\n    \u222b (t : \u211d) in Ioi 0, a * hh b (t / c) = a * c / b * \u03c0 :=\n  hh_integrable_aux ha hb hc |>.2\n\nlemma hh_integral' : \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t = 2 * \u03c0 ^ 2 := by\n  have := hh_integral (a := 1) (b := 1 / (2 * \u03c0)) (c := 1) (by positivity) (by positivity) (by positivity)\n  convert this using 1 <;> simp ; ring\n\nlemma bound_sum_log {C : \u211d} (hf0 : f 0 = 0) (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let ggg (i : \u2115) : \u211d := if i = 0 then 1 else gg x i\n\n  have l0 : x \u2260 0 := by linarith\n  have l1 i : 0 \u2264 ggg i := by by_cases hi : i = 0 <;> simp [ggg, hi, gg] ; positivity\n  have l2 : Antitone ggg := by\n    intro i j hij ; by_cases hi : i = 0 <;> by_cases hj : j = 0 <;> simp [ggg, hi, hj]\n    \u00b7 exact gg_le_one _\n    \u00b7 omega\n    \u00b7 simp only [gg_of_hh l0]\n      gcongr\n      apply hh_antitone one_div_two_pi_mem_Ioo\n      \u00b7 simp ; positivity\n      \u00b7 simp ; positivity\n      \u00b7 gcongr\n  have l3 : 0 \u2264 C := by simpa [cumsum, hf0] using hf 1\n\n  have l4 : 0 \u2264 \u222b (t : \u211d) in Ioi 0, hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) t :=\n    set_integral_nonneg measurableSet_Ioi (fun x hx => hh_nonneg _ (LT.lt.le hx))\n\n  have l5 {n : \u2115} : AntitoneOn (fun t \u21a6 x\u207b\u00b9 * hh (1 / (2 * \u03c0)) (t / x)) (Ioc 0 n) := by\n    intro u \u27e8hu1, _\u27e9 v \u27e8hv1, _\u27e9 huv\n    simp only\n    apply mul_le_mul le_rfl ?_ (hh_nonneg _ (by positivity)) (by positivity)\n    apply hh_antitone one_div_two_pi_mem_Ioo (by simp ; positivity) (by simp ; positivity)\n    apply (div_le_div_right (by positivity)).mpr huv\n\n  have l6 {n : \u2115} : IntegrableOn (fun t \u21a6 x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (Icc 0 n) volume := by\n    apply IntegrableOn.mono_set (hh_integrable (by positivity) (by positivity) (by positivity)) Icc_subset_Ici_self\n\n  apply Real.tsum_le_of_sum_range_le (fun n => by positivity) ; intro n\n  convert_to \u2211 i in Finset.range n, \u2016f i\u2016 * ggg i \u2264 _\n  \u00b7 congr ; ext i\n    by_cases hi : i = 0\n    \u00b7 simp [hi, hf0]\n    \u00b7 field_simp [hi, ggg, gg]\n\n  apply cancel_main' (fun _ => norm_nonneg _) (by simp [hf0]) l1 hf l2 n |>.trans\n  gcongr ; simp [ggg, cumsum, gg_of_hh l0]\n\n  by_cases hn : n = 0 ; simp [hn] ; positivity\n  replace hn : 0 < n := by omega\n  have : Finset.range n = {0} \u222a Finset.Ico 1 n := by\n    ext i ; simp ; by_cases hi : i = 0 <;> simp [hi, hn] ; omega\n  simp [this, Finset.sum_union]\n  convert_to \u2211 x_1 in Finset.Ico 1 n, x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (\u2191x_1 / x) \u2264 _\n  \u00b7 apply Finset.sum_congr rfl (fun i hi => ?_)\n    simp at hi\n    have : i \u2260 0 := by omega\n    simp [this]\n  simp_rw [Finset.sum_Ico_eq_sum_range, add_comm 1]\n  have := @sum_le_integral 0 (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t / x)) (n - 1) (by simpa using l5) (by simpa using l6)\n  simp only [zero_add] at this\n  apply this.trans\n  rw [@intervalIntegral.integral_comp_div \u211d _ _ 0 \u2191(n - 1) x (fun t => x\u207b\u00b9 * hh (\u03c0\u207b\u00b9 * 2\u207b\u00b9) (t)) l0]\n  simp [\u2190 mul_assoc, mul_inv_cancel l0]\n  have : (0 : \u211d) \u2264 \u2191(n - 1) / x := by positivity\n  rw [intervalIntegral.intervalIntegral_eq_integral_uIoc]\n  simp [this]\n  apply integral_mono_measure\n  \u00b7 apply Measure.restrict_mono Ioc_subset_Ioi_self le_rfl\n  \u00b7 apply eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi)\n    intro x (hx : 0 < x)\n    apply hh_nonneg _ hx.le\n  \u00b7 have := (@hh_integrable 1 (1 / (2 * \u03c0)) 1 (by positivity) (by positivity) (by positivity))\n    simpa using this.mono_set Ioi_subset_Ici_self\n\nlemma bound_sum_log0 {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + \u222b t in Ioi 0, hh (1 / (2 * \u03c0)) t) := by\n\n  let f0 i := if i = 0 then 0 else f i\n  have l1 : chebyWith C f0 := by\n    intro n ; refine Finset.sum_le_sum (fun i _ => ?_) |>.trans (hf n)\n    by_cases hi : i = 0 <;> simp [hi, f0]\n  have l2 i : \u2016f i\u2016 / i = \u2016f0 i\u2016 / i := by by_cases hi : i = 0 <;> simp [hi, f0]\n  simp_rw [l2] ; apply bound_sum_log rfl l1 hx\n\nlemma bound_sum_log' {C : \u211d} (hf : chebyWith C f) {x : \u211d} (hx : 1 \u2264 x) :\n    \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 \u2264 C * (1 + 2 * \u03c0 ^ 2) := by\n  simpa only [hh_integral'] using bound_sum_log0 hf hx\n\nlemma summable_fourier (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  exact Summable.of_nonneg_of_le (fun _ => norm_nonneg _) l6 (by simpa using l5.const_smul (W21.norm \u03c8))\n\nlemma bound_I1 (x : \u211d) (hx : 0 < x) (\u03c8 : W21) (hcheby : cheby f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264\n    W21.norm \u03c8 \u2022 \u2211' i, \u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9 := by\n\n  have l5 : Summable fun i \u21a6 \u2016f i\u2016 / \u2191i * ((1 + (1 / (2 * \u2191\u03c0) * \u2191(Real.log (\u2191i / x))) ^ 2)\u207b\u00b9) := by\n    simpa using limiting_fourier_lim1_aux hcheby hx 1 zero_le_one\n  have l6 i : \u2016f i / i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (i / x))\u2016 \u2264\n      W21.norm \u03c8 * (\u2016f i\u2016 / i * (1 + (1 / (2 * \u03c0) * log (i / x)) ^ 2)\u207b\u00b9) := by\n    convert mul_le_mul_of_nonneg_left (decay_bounds_key \u03c8 (1 / (2 * \u03c0) * log (i / x))) (norm_nonneg (f i / i)) using 1\n    \u00b7 simp [norm_mul]\n    \u00b7 change _ = _ * (W21.norm \u03c8 * _) ; simp [W21.norm] ; ring\n  have l1 : Summable fun i \u21a6 \u2016f i / \u2191i * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191i / x))\u2016 := by\n    exact summable_fourier x hx \u03c8 hcheby\n  apply (norm_tsum_le_tsum_norm l1).trans\n  simpa only [\u2190 tsum_const_smul _ l5] using tsum_mono l1 (by simpa using l5.const_smul (W21.norm \u03c8)) l6\n\nlemma bound_I1' {C : \u211d} (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21) (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))\u2016 \u2264 W21.norm \u03c8 * C * (1 + 2 * \u03c0 ^ 2) := by\n\n  apply bound_I1 x (by linarith) \u03c8 \u27e8_, hcheby\u27e9 |>.trans\n  rw [smul_eq_mul, mul_assoc]\n  apply mul_le_mul le_rfl (bound_sum_log' hcheby hx) ?_ W21.norm_nonneg\n  apply tsum_nonneg (fun i => by positivity)\n\nlemma bound_I2 (x : \u211d) (\u03c8 : W21) :\n    \u2016\u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (2 * \u03c0 ^ 2) := by\n\n  have key a : \u2016\ud835\udcd5 \u03c8 (a / (2 * \u03c0))\u2016 \u2264 W21.norm \u03c8 * (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := decay_bounds_key \u03c8 _\n  have twopi : 0 \u2264 2 * \u03c0 := by simp [pi_nonneg]\n  have l3 : Integrable (fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.comp_div (by norm_num [pi_ne_zero])\n  have l2 : IntegrableOn (fun i \u21a6 W21.norm \u03c8 * (1 + (i / (2 * \u03c0)) ^ 2)\u207b\u00b9) (Ici (-Real.log x)) := by\n    exact (l3.const_mul _).integrableOn\n  have l1 : IntegrableOn (fun i \u21a6 \u2016\ud835\udcd5 \u03c8 (i / (2 * \u03c0))\u2016) (Ici (-Real.log x)) := by\n    refine ((l3.const_mul (W21.norm \u03c8)).mono' ?_ ?_).integrableOn\n    \u00b7 apply Continuous.aestronglyMeasurable ; continuity\n    \u00b7 simp only [norm_norm, key] ; simp\n  have l5 : 0 \u2264\u1d50[volume] fun a \u21a6 (1 + (a / (2 * \u03c0)) ^ 2)\u207b\u00b9 := by apply eventually_of_forall ; intro x ; positivity\n  refine (norm_integral_le_integral_norm _).trans <| (set_integral_mono l1 l2 key).trans ?_\n  rw [integral_mul_left] ; gcongr ; apply W21.norm_nonneg\n  refine (set_integral_le_integral l3 l5).trans ?_\n  rw [Measure.integral_comp_div (fun x => (1 + x ^ 2)\u207b\u00b9) (2 * \u03c0)]\n  simp [abs_eq_self.mpr twopi] ; ring_nf ; rfl\n\nlemma bound_main {C : \u211d} (A : \u2102) (x : \u211d) (hx : 1 \u2264 x) (\u03c8 : W21)\n    (hcheby : chebyWith C f) :\n    \u2016\u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\u2016 \u2264\n      W21.norm \u03c8 * (C * (1 + 2 * \u03c0 ^ 2) + \u2016A\u2016 * (2 * \u03c0 ^ 2)) := by\n\n  have l1 := bound_I1' x hx \u03c8 hcheby\n  have l2 := mul_le_mul (le_refl \u2016A\u2016) (bound_I2 x \u03c8) (by positivity) (by positivity)\n  apply norm_sub_le _ _ |>.trans ; rw [norm_mul]\n  convert _root_.add_le_add l1 l2 using 1 ; ring\n\n/-%%\n\\begin{lemma}[Limiting identity for Schwartz functions]\\label{schwarz-id}\\lean{limiting_cor_schwartz}\\leanok  The previous corollary also holds for functions $\\psi$ that are assumed to be in the Schwartz class, as opposed to being $C^2$ and compactly supported.\n\\end{lemma}\n%%-/\n\nlemma limiting_cor_W21 (\u03c8 : W21) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) := by\n\n  -- Shorter notation for clarity\n  let S1 x (\u03c8 : \u211d \u2192 \u2102) := \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * Real.log (\u2191n / x))\n  let S2 x (\u03c8 : \u211d \u2192 \u2102) := \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))\n  let S x \u03c8 := S1 x \u03c8 - S2 x \u03c8 ; change Tendsto (fun x \u21a6 S x \u03c8) atTop (\ud835\udcdd 0)\n\n  -- Build the truncation\n  obtain g := exists_trunc\n  let \u03a8 R := g.scale R * \u03c8\n  have key R : Tendsto (fun x \u21a6 S x (\u03a8 R)) atTop (\ud835\udcdd 0) := limiting_cor (\u03a8 R) hf hcheby hG hG'\n\n  -- Choose the truncation radius\n  obtain \u27e8C, hcheby\u27e9 := hcheby\n  have hC : 0 \u2264 C := by\n    have : \u2016f 0\u2016 \u2264 C := by simpa [cumsum] using hcheby 1\n    have : 0 \u2264 \u2016f 0\u2016 := by positivity\n    linarith\n  have key2 : Tendsto (fun R \u21a6 W21.norm (\u03c8 - \u03a8 R)) atTop (\ud835\udcdd 0) := W21_approximation \u03c8 g\n  simp_rw [Metric.tendsto_nhds] at key key2 \u22a2 ; intro \u03b5 h\u03b5\n  let M := C * (1 + 2 * \u03c0 ^ 2) + \u2016(A : \u2102)\u2016 * (2 * \u03c0 ^ 2)\n  obtain \u27e8R, hR\u03c8\u27e9 := (key2 ((\u03b5 / 2) / (1 + M)) (by positivity)).exists\n  simp only [dist_zero_right, Real.norm_eq_abs, abs_eq_self.mpr W21.norm_nonneg] at hR\u03c8 key\n\n  -- Apply the compact support case\n  filter_upwards [eventually_ge_atTop 1, key R (\u03b5 / 2) (by positivity)] with x hx key\n\n  -- Control the tail term\n  have key3 : \u2016S x (\u03c8 - \u03a8 R)\u2016 < \u03b5 / 2 := by\n    have : \u2016S x _\u2016 \u2264 _ * M := @bound_main f C A x hx (\u03c8 - \u03a8 R) hcheby\n    apply this.trans_lt\n    apply (mul_le_mul (d := 1 + M) le_rfl (by simp) (by positivity) W21.norm_nonneg).trans_lt\n    have : 0 < 1 + M := by positivity\n    convert (mul_lt_mul_right this).mpr hR\u03c8 using 1 ; field_simp ; ring\n\n  -- Conclude the proof\n  have S1_sub_1 x : \ud835\udcd5 (\u21d1\u03c8 - \u21d1(\u03a8 R)) x = \ud835\udcd5 \u03c8 x - \ud835\udcd5 (\u03a8 R) x := by\n    have l1 : AEStronglyMeasurable (fun x_1 : \u211d \u21a6 cexp (-(2 * \u2191\u03c0 * (\u2191x_1 * \u2191x) * I))) volume := by\n      refine (Continuous.mul ?_ continuous_const).neg.cexp.aestronglyMeasurable\n      apply continuous_const.mul <| contDiff_ofReal.continuous.mul continuous_const\n    simp [Real.fourierIntegral_eq', mul_sub] ; apply integral_sub\n    \u00b7 apply \u03c8.hf.bdd_mul l1 ; use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n    \u00b7 apply (\u03a8 R : W21) |>.hf |>.bdd_mul l1\n      use 1 ; simp [Complex.norm_eq_abs, Complex.abs_exp]\n\n  have S1_sub : S1 x (\u03c8 - \u03a8 R) = S1 x \u03c8 - S1 x (\u03a8 R) := by\n    simp [S1, S1_sub_1, mul_sub] ; apply tsum_sub\n    \u00b7 have := summable_fourier x (by positivity) \u03c8 \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n    \u00b7 have := summable_fourier x (by positivity) (\u03a8 R) \u27e8_, hcheby\u27e9\n      rw [summable_norm_iff] at this\n      simpa using this\n\n  have S2_sub : S2 x (\u03c8 - \u03a8 R) = S2 x \u03c8 - S2 x (\u03a8 R) := by\n    simp [S2, S1_sub_1] ; rw [integral_sub] ; ring\n    \u00b7 exact \u03c8.integrable_fourier (by positivity) |>.restrict\n    \u00b7 exact (\u03a8 R : W21).integrable_fourier (by positivity) |>.restrict\n\n  have S_sub : S x (\u03c8 - \u03a8 R) = S x \u03c8 - S x (\u03a8 R) := by simp [S, S1_sub, S2_sub] ; ring\n  simpa [S_sub, \u03a8] using norm_add_le _ _ |>.trans_lt (_root_.add_lt_add key3 key)\n\nlemma limiting_cor_schwartz (\u03c8 : \ud835\udce2(\u211d, \u2102)) (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re}) :\n    Tendsto (fun x : \u211d \u21a6 \u2211' n, f n / n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n      A * \u222b u in Set.Ici (-log x), \ud835\udcd5 \u03c8 (u / (2 * \u03c0))) atTop (\ud835\udcdd 0) :=\n  limiting_cor_W21 \u03c8 hf hcheby hG hG'\n\n/-%%\n\\begin{proof}\n\\uses{limiting-cor, smooth-ury}\\leanok\nFor any $R>1$, one can use a smooth cutoff function (provided by Lemma \\ref{smooth-ury} to write $\\psi = \\psi_{\\leq R} + \\psi_{>R}$, where $\\psi_{\\leq R}$ is $C^2$ (in fact smooth) and compactly supported (on $[-R,R]$), and $\\psi_{>R}$ obeys bounds of the form\n$$ |\\psi_{>R}(t)|, |\\psi''_{>R}(t)| \\ll R^{-1} / (1 + |t|^2) $$\nwhere the implied constants depend on $\\psi$.  By Lemma \\ref{decay} we then have\n$$ \\hat \\psi_{>R}(u) \\ll R^{-1} / (1+|u|^2).$$\nUsing this and \\eqref{cheby} one can show that\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{>R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ), A \\int_{-\\infty}^\\infty \\hat \\psi_{>R} (\\frac{u}{2\\pi})\\ du \\ll R^{-1} $$\n(with implied constants also depending on $A$), while from Lemma \\ref{limiting-cor} one has\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi_{\\leq R}( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi_{\\leq R} (\\frac{u}{2\\pi})\\ du + o(1).$$\nCombining the two estimates and letting $R$ be large, we obtain the claim.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Bijectivity of Fourier transform]\\label{bij}\\lean{fourier_surjection_on_schwartz}\\leanok  The Fourier transform is a bijection on the Schwartz class.\n\\end{lemma}\n%%-/\n\n-- just the surjectivity is stated here, as this is all that is needed for the current application, but perhaps one should state and prove bijectivity instead\n\nlemma fourier_surjection_on_schwartz (f : \ud835\udce2(\u211d, \u2102)) : \u2203 g : \ud835\udce2(\u211d, \u2102), \ud835\udcd5 g = f := by\n  use FS (FS (FS f)) ; ext x ; nth_rewrite 2 [\u2190 FS4 f] ; simp\n\n/-%%\n\\begin{proof}\n  \\leanok\n This is a standard result in Fourier analysis.\nIt can be proved here by appealing to Mellin inversion, Theorem \\ref{MellinInversion}.\nIn particular, given $f$ in the Schwartz class, let $F : \\R_+ \\to \\C : x \\mapsto f(\\log x)$ be a function in the ``Mellin space''; then the Mellin transform of $F$ on the imaginary axis $s=it$ is the Fourier transform of $f$.  The Mellin inversion theorem gives Fourier inversion.\n\\end{proof}\n%%-/\n\ndef toSchwartz (f : \u211d \u2192 \u2102) (h1 : ContDiff \u211d \u22a4 f) (h2 : HasCompactSupport f) : \ud835\udce2(\u211d, \u2102) where\n  toFun := f\n  smooth' := h1\n  decay' k n := by\n    have l1 : Continuous (fun x => \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := by\n      have : ContDiff \u211d \u22a4 (iteratedFDeriv \u211d n f) := h1.iteratedFDeriv_right le_top\n      exact Continuous.mul (by continuity) this.continuous.norm\n    have l2 : HasCompactSupport (fun x \u21a6 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f x\u2016) := (h2.iteratedFDeriv _).norm.mul_left\n    simpa using l1.bounded_above_of_compact_support l2\n\n@[simp] lemma toSchwartz_apply (f : \u211d \u2192 \u2102) {h1 h2 x} : SchwartzMap.mk f h1 h2 x = f x := rfl\n\nlemma comp_exp_support0 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in \ud835\udcdd 0, \u03a8 x = 0 :=\n  not_mem_tsupport_iff_eventuallyEq.mp (fun h => lt_irrefl 0 <| mem_Ioi.mp (hplus h))\n\nlemma comp_exp_support1 {\u03a8 : \u211d \u2192 \u2102} (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    \u2200\u1da0 x in atBot, \u03a8 (exp x) = 0 :=\n  Real.tendsto_exp_atBot <| comp_exp_support0 hplus\n\nlemma comp_exp_support2 {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) :\n    \u2200\u1da0 (x : \u211d) in atTop, (\u03a8 \u2218 rexp) x = 0 := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop] at hsupp\n  exact Real.tendsto_exp_atTop hsupp.2\n\ntheorem comp_exp_support {\u03a8 : \u211d \u2192 \u2102} (hsupp : HasCompactSupport \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    HasCompactSupport (\u03a8 \u2218 rexp) := by\n  simp only [hasCompactSupport_iff_eventuallyEq, coclosedCompact_eq_cocompact, cocompact_eq_atBot_atTop]\n  exact \u27e8comp_exp_support1 hplus, comp_exp_support2 hsupp\u27e9\n\nlemma wiener_ikehara_smooth_aux (l0 : Continuous \u03a8) (hsupp : HasCompactSupport \u03a8)\n    (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) (x : \u211d) (hx : 0 < x) :\n    \u222b (u : \u211d) in Ioi (-Real.log x), \u2191(rexp u) * \u03a8 (rexp u) = \u222b (y : \u211d) in Ioi (1 / x), \u03a8 y := by\n\n  have l1 : ContinuousOn rexp (Ici (-Real.log x)) := by fun_prop\n  have l2 : Tendsto rexp atTop atTop := Real.tendsto_exp_atTop\n  have l3 t (_ : t \u2208 Ioi (-log x)) : HasDerivWithinAt rexp (rexp t) (Ioi t) t :=\n    (Real.hasDerivAt_exp t).hasDerivWithinAt\n  have l4 : ContinuousOn \u03a8 (rexp '' Ioi (-Real.log x)) := by fun_prop\n  have l5 : IntegrableOn \u03a8 (rexp '' Ici (-Real.log x)) volume :=\n    (l0.integrable_of_hasCompactSupport hsupp).integrableOn\n  have l6 : IntegrableOn (fun x \u21a6 rexp x \u2022 (\u03a8 \u2218 rexp) x) (Ici (-Real.log x)) volume := by\n    refine (Continuous.integrable_of_hasCompactSupport (by continuity) ?_).integrableOn\n    change HasCompactSupport (rexp \u2022 (\u03a8 \u2218 rexp))\n    exact (comp_exp_support hsupp hplus).smul_left\n  have := MeasureTheory.integral_comp_smul_deriv_Ioi l1 l2 l3 l4 l5 l6\n  simpa [Real.exp_neg, Real.exp_log hx] using this\n\ntheorem wiener_ikehara_smooth_sub (h1 : Integrable \u03a8) (hplus : closure (Function.support \u03a8) \u2286 Ioi 0) :\n    Tendsto (fun x \u21a6 (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n\n  obtain \u27e8\u03b5, h\u03b5, hh\u27e9 := Metric.eventually_nhds_iff.mp <| comp_exp_support0 hplus\n  apply tendsto_nhds_of_eventually_eq ; filter_upwards [eventually_gt_atTop \u03b5\u207b\u00b9] with x hx\u03b5\n\n  have l1 : Integrable (indicator (Ioi x\u207b\u00b9) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n  have l2 : Integrable (indicator (Ioi 0) (fun x : \u211d => \u03a8 x)) := h1.indicator measurableSet_Ioi\n\n  simp_rw [\u2190 MeasureTheory.integral_indicator measurableSet_Ioi, \u2190 mul_sub, \u2190 integral_sub l1 l2]\n  simp ; right ; apply MeasureTheory.integral_eq_zero_of_ae ; apply eventually_of_forall ; intro t ; simp\n\n  have h\u03b5' : 0 < \u03b5\u207b\u00b9 := by positivity\n  have hx : 0 < x := by linarith\n  have hx' : 0 < x\u207b\u00b9 := by positivity\n  have h\u03b5x : x\u207b\u00b9 < \u03b5 := by apply (inv_lt h\u03b5 hx).mp hx\u03b5\n\n  have l3 : Ioi 0 = Ioc 0 x\u207b\u00b9 \u222a Ioi x\u207b\u00b9 := by\n    ext t ; simp ; constructor <;> intro h\n    \u00b7 simp [h, le_or_lt]\n    \u00b7 cases h <;> linarith\n  have l4 : Disjoint (Ioc 0 x\u207b\u00b9) (Ioi x\u207b\u00b9) := by simp\n  have l5 := Set.indicator_union_of_disjoint l4 \u03a8\n  rw [l3, l5] ; ring_nf\n  by_cases ht : t \u2208 Ioc 0 x\u207b\u00b9 <;> simp [ht]\n  apply hh ; simp at ht \u22a2\n  have : |t| \u2264 x\u207b\u00b9 := by rw [abs_le] ; constructor <;> linarith\n  linarith\n\n/-%%\n\\begin{corollary}[Smoothed Wiener-Ikehara]\\label{WienerIkeharaSmooth}\\lean{wiener_ikehara_smooth}\\leanok\n  If $\\Psi: (0,\\infty) \\to \\C$ is smooth and compactly supported away from the origin, then, then\n$$ \\sum_{n=1}^\\infty f(n) \\Psi( \\frac{n}{x} ) = A x \\int_0^\\infty \\Psi(y)\\ dy + o(x)$$\nas $u \\to \\infty$.\n\\end{corollary}\n%%-/\n\nlemma wiener_ikehara_smooth (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x - A * \u222b y in Set.Ioi 0, \u03a8 y) atTop (nhds 0) := by\n\n  let h (x : \u211d) : \u2102 := rexp (2 * \u03c0 * x) * \u03a8 (exp (2 * \u03c0 * x))\n  have h1 : ContDiff \u211d \u22a4 h := by\n    have : ContDiff \u211d \u22a4 (fun x : \u211d => (rexp (2 * \u03c0 * x))) := (contDiff_const.mul contDiff_id).exp\n    exact (contDiff_ofReal.comp this).mul (hsmooth.comp this)\n  have h2 : HasCompactSupport h := by\n    have : 2 * \u03c0 \u2260 0 := by simp [pi_ne_zero]\n    simpa using (comp_exp_support hsupp hplus).comp_smul this |>.mul_left\n  obtain \u27e8g, hg\u27e9 := fourier_surjection_on_schwartz (toSchwartz h h1 h2)\n\n  have why (x : \u211d) : 2 * \u03c0 * x / (2 * \u03c0) = x := by field_simp ; ring\n  have l1 {y} (hy : 0 < y) : y * \u03a8 y = \ud835\udcd5 g (1 / (2 * \u03c0) * Real.log y) := by\n    field_simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [Real.exp_log hy]\n\n  have key := limiting_cor_schwartz g hf hcheby hG hG'\n\n  have l2 : \u2200\u1da0 x in atTop, \u2211' (n : \u2115), f n / \u2191n * \ud835\udcd5 (\u21d1g) (1 / (2 * \u03c0) * Real.log (\u2191n / x)) =\n      \u2211' (n : \u2115), f n * \u03a8 (\u2191n / x) / x := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr ; ext n\n    by_cases hn : n = 0 ; simp [hn, (comp_exp_support0 hplus).self_of_nhds]\n    rw [\u2190 l1 (by positivity)]\n    have : (n : \u2102) \u2260 0 := by simpa using hn\n    have : (x : \u2102) \u2260 0 := by simpa using hx.ne.symm\n    field_simp ; ring\n\n  have l3 : \u2200\u1da0 x in atTop, \u2191A * \u222b (u : \u211d) in Ici (-Real.log x), \ud835\udcd5 (\u21d1g) (u / (2 * \u03c0)) =\n      \u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y := by\n    filter_upwards [eventually_gt_atTop 0] with x hx\n    congr 1 ; simp [hg, toSchwartz, h] ; norm_cast ; field_simp [why] ; norm_cast\n    rw [MeasureTheory.integral_Ici_eq_integral_Ioi]\n    exact wiener_ikehara_smooth_aux hsmooth.continuous hsupp hplus x hx\n\n  have l4 : Tendsto (fun x => (\u2191A * \u222b (y : \u211d) in Ioi x\u207b\u00b9, \u03a8 y) - \u2191A * \u222b (y : \u211d) in Ioi 0, \u03a8 y) atTop (\ud835\udcdd 0) := by\n    exact wiener_ikehara_smooth_sub (hsmooth.continuous.integrable_of_hasCompactSupport hsupp) hplus\n\n  simpa [tsum_div_const] using (key.congr' <| EventuallyEq.sub l2 l3) |>.add l4\n\n/-%%\n\\begin{proof}\n\\uses{bij,schwarz-id}\\leanok\n By Lemma \\ref{bij}, we can write\n$$ y \\Psi(y) = \\hat \\psi( \\frac{1}{2\\pi} \\log y )$$\nfor all $y>0$ and some Schwartz function $\\psi$.  Making this substitution, the claim is then equivalent after standard manipulations to\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = A \\int_{-\\infty}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du + o(1)$$\nand the claim follows from Lemma \\ref{schwarz-id}.\n\\end{proof}\n%%-/\n\nlemma wiener_ikehara_smooth' (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcheby : cheby f)\n    (hG: ContinuousOn G {s | 1 \u2264 s.re}) (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) :=\n  tendsto_sub_nhds_zero_iff.mp <| wiener_ikehara_smooth hf hcheby hG hG' hsmooth hsupp hplus\n\nlocal instance {E : Type*} : Coe (E \u2192 \u211d) (E \u2192 \u2102) := \u27e8fun f n => f n\u27e9\n\n@[norm_cast]\ntheorem set_integral_ofReal {f : \u211d \u2192 \u211d} {s : Set \u211d} : \u222b x in s, (f x : \u2102) = \u222b x in s, f x :=\n  integral_ofReal\n\nlemma wiener_ikehara_smooth_real {f : \u2115 \u2192 \u211d} {\u03a8 : \u211d \u2192 \u211d} (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3'))\n    (hcheby : cheby f) (hG: ContinuousOn G {s | 1 \u2264 s.re})\n    (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hsmooth: ContDiff \u211d \u22a4 \u03a8) (hsupp: HasCompactSupport \u03a8) (hplus: closure (Function.support \u03a8) \u2286 Set.Ioi 0) :\n    Tendsto (fun x : \u211d \u21a6 (\u2211' n, f n * \u03a8 (n / x)) / x) atTop (nhds (A * \u222b y in Set.Ioi 0, \u03a8 y)) := by\n\n  let \u03a8' := ofReal' \u2218 \u03a8\n  have l1 : ContDiff \u211d \u22a4 \u03a8' := contDiff_ofReal.comp hsmooth\n  have l2 : HasCompactSupport \u03a8' := hsupp.comp_left rfl\n  have l3 : closure (Function.support \u03a8') \u2286 Ioi 0 := by rwa [Function.support_comp_eq] ; simp\n  have key := (continuous_re.tendsto _).comp (@wiener_ikehara_smooth' A \u03a8 G f hf hcheby hG hG' l1 l2 l3)\n  simp at key ; norm_cast at key\n\nlemma interval_approx_inf (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      \u03c8 \u2264 indicator (Ico a b) 1 \u2227 b - a - \u03b5 \u2264 \u222b y in Ioi 0, \u03c8 y := by\n\n  have l1 : Iio ((b - a) / 3) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < (b - a) / 3)\n  have l2 : a < a + \u03b5 / 2 := by linarith\n  have l3 : b - \u03b5 / 2 < b := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 simp [h5, hab.ne, Icc_subset_Ioi_iff hab.le, ha]\n  \u00b7 exact h4.trans <| indicator_le_indicator_of_subset Ioo_subset_Ico_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a - \u03b5 := by linarith\n    have l5 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Icc (a + \u03b5 / 2) (b - \u03b5 / 2) \u2229 Ioi 0 = Icc (a + \u03b5 / 2) (b - \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Icc (a + \u03b5 / 2) (b - \u03b5 / 2)) 1 y = b - a - \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; apply set_integral_mono ?_ l8 h3\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Icc]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\nlemma interval_approx_sup (ha : 0 < a) (hab : a < b) :\n    \u2200\u1da0 \u03b5 in \ud835\udcdd[>] 0, \u2203 \u03c8 : \u211d \u2192 \u211d, ContDiff \u211d \u22a4 \u03c8 \u2227 HasCompactSupport \u03c8 \u2227 closure (Function.support \u03c8) \u2286 Set.Ioi 0 \u2227\n      indicator (Ico a b) 1 \u2264 \u03c8 \u2227 \u222b y in Ioi 0, \u03c8 y \u2264 b - a + \u03b5 := by\n\n  have l1 : Iio (a / 2) \u2208 \ud835\udcdd[>] 0 := nhdsWithin_le_nhds <| Iio_mem_nhds (by linarith)\n  filter_upwards [self_mem_nhdsWithin, l1] with \u03b5 (h\u03b5 : 0 < \u03b5) (h\u03b5' : \u03b5 < a / 2)\n  have l2 : a - \u03b5 / 2 < a := by linarith\n  have l3 : b < b + \u03b5 / 2 := by linarith\n  obtain \u27e8\u03c8, h1, h2, h3, h4, h5\u27e9 := smooth_urysohn_support_Ioo l2 l3\n  refine \u27e8\u03c8, h1, h2, ?_, ?_, ?_\u27e9\n  \u00b7 have l4 : a - \u03b5 / 2 < b + \u03b5 / 2 := by linarith\n    have l5 : \u03b5 / 2 < a := by linarith\n    simp [h5, l4.ne, Icc_subset_Ioi_iff l4.le, l5]\n  \u00b7 apply le_trans ?_ h3\n    apply indicator_le_indicator_of_subset Ico_subset_Icc_self (by simp)\n  \u00b7 have l4 : 0 \u2264 b - a + \u03b5 := by linarith\n    have l5 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2286 Ioi 0 := by intro t ht ; simp at ht \u22a2 ; linarith\n    have l6 : Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) \u2229 Ioi 0 = Ioo (a - \u03b5 / 2) (b + \u03b5 / 2) := inter_eq_left.mpr l5\n    have l7 : \u222b y in Ioi 0, indicator (Ioo (a - \u03b5 / 2) (b + \u03b5 / 2)) 1 y = b - a + \u03b5 := by\n      simp [l6] ; convert ENNReal.toReal_ofReal l4 using 3 ; ring\n    have l8 : IntegrableOn \u03c8 (Ioi 0) volume := (h1.continuous.integrable_of_hasCompactSupport h2).integrableOn\n    rw [\u2190 l7] ; refine set_integral_mono l8 ?_ h4\n    rw [IntegrableOn, integrable_indicator_iff measurableSet_Ioo]\n    apply IntegrableOn.mono ?_ subset_rfl Measure.restrict_le_self\n    apply integrableOn_const.mpr\n    simp\n\n", "theoremStatement": "lemma WI_summable {f : \u2115 \u2192 \u211d} {g : \u211d \u2192 \u211d} (hg : HasCompactSupport g) (hx : 0 < x) :\n    Summable (fun n => f n * g (n / x)) ", "theoremName": "WI_summable", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "98c2a8322b6376a3412e2f0fbb911204c36f9c04", "date": "2024-04-04"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 1826, "tokenPositionInFile": 91696, "theoremPositionInFile": 144}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 110, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", 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"Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8M, hM\u27e9 := hg.bddAbove.mono subset_closure\n  apply summable_of_finite_support\n  simp ; apply Finite.inter_of_right ; rw [finite_iff_bddAbove]\n  exact \u27e8Nat.ceil (M * x), fun i hi => by simpa using Nat.ceil_mono ((div_le_iff hx).mp (hM hi))\u27e9", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 254}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\nlemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.TendstoAtTop_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f atTop (\ud835\udcdd 0) := by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioi_mem_atTop b] with c hc; rw [mem_Ioi] at hc\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp\n\nlemma Filter.BigO_zero_atZero_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[\ud835\udcdd[>] 0] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < a)] with c hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h, (mem_Ioo.mp hc).2]\n\nlemma Filter.BigO_zero_atTop_of_support_in_Icc {a b : \u211d} (f : \u211d \u2192 \ud835\udd42)\n    (fSupp : f.support \u2286 Set.Icc a b):\n    f =O[atTop] fun _ \u21a6 (0 : \u211d) := by\n  refine Eventually.isBigO ?_\n  filter_upwards [Ioi_mem_atTop b] with c hc; replace hc := mem_Ioi.mp hc\n  refine norm_le_zero_iff.mpr <| Function.support_subset_iff'.mp fSupp c ?_\n  exact fun h \u21a6 by linarith [mem_Icc.mp h]\n\n-- steal coerction lemmas from EulerProducts.Auxiliary because of build issues, and add new ones\nnamespace Complex\n-- see https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Differentiability.20of.20the.20natural.20map.20.E2.84.9D.20.E2.86.92.20.E2.84.82/near/418095234\n\nlemma hasDerivAt_ofReal (x : \u211d) : HasDerivAt ofReal' 1 x :=\n  HasDerivAt.ofReal_comp <| hasDerivAt_id x\n\nlemma deriv_ofReal (x : \u211d) : deriv ofReal' x = 1 :=\n  (hasDerivAt_ofReal x).deriv\n\nlemma differentiableAt_ofReal (x : \u211d) : DifferentiableAt \u211d ofReal' x :=\n  (hasDerivAt_ofReal x).differentiableAt\n\nlemma differentiable_ofReal : Differentiable \u211d ofReal' :=\n  ofRealCLM.differentiable\n\nend Complex\n\nlemma DifferentiableAt.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    DifferentiableAt \u211d (fun x : \u211d \u21a6 e x) z :=\n  hf.hasDerivAt.comp_ofReal.differentiableAt\n\nlemma Differentiable.comp_ofReal {e : \u2102 \u2192 \u2102} (h : Differentiable \u2102 e) :\n    Differentiable \u211d (fun x : \u211d \u21a6 e x) :=\n  fun _ \u21a6 h.differentiableAt.comp_ofReal\n\nlemma DifferentiableAt.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} (hf : DifferentiableAt \u211d f z) :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z :=\n  hf.hasDerivAt.ofReal_comp.differentiableAt\n\nlemma Differentiable.ofReal_comp {f : \u211d \u2192 \u211d} (hf : Differentiable \u211d f) :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) :=\n  fun _ \u21a6 hf.differentiableAt.ofReal_comp\n\nopen Complex ContinuousLinearMap in\nlemma HasDerivAt.of_hasDerivAt_ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} {u : \u2102}\n    (hf : HasDerivAt (fun y \u21a6 (f y : \u2102)) u z) :\n    \u2203 u' : \u211d, u = u' \u2227 HasDerivAt f u' z := by\n  lift u to \u211d\n  \u00b7 have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv\n    simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H\n    rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H\n  refine \u27e8u, rfl, ?_\u27e9\n  convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt\n  rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]\n\nlemma DifferentiableAt.ofReal_comp_iff {z : \u211d} {f : \u211d \u2192 \u211d} :\n    DifferentiableAt \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) z \u2194 DifferentiableAt \u211d f z := by\n  refine \u27e8fun H \u21a6 ?_, ofReal_comp\u27e9\n  obtain \u27e8u, _, hu\u2082\u27e9 := H.hasDerivAt.of_hasDerivAt_ofReal_comp\n  exact HasDerivAt.differentiableAt hu\u2082\n\nlemma Differentiable.ofReal_comp_iff {f : \u211d \u2192 \u211d} :\n    Differentiable \u211d (fun (y : \u211d) \u21a6 (f y : \u2102)) \u2194 Differentiable \u211d f :=\n  forall_congr' fun _ \u21a6 DifferentiableAt.ofReal_comp_iff\n\nlemma deriv.ofReal_comp {z : \u211d} {f : \u211d \u2192 \u211d} :\n    deriv (fun (y : \u211d) \u21a6 (f y : \u2102)) z = deriv f z := by\n  by_cases hf : DifferentiableAt \u211d f z\n  \u00b7 exact hf.hasDerivAt.ofReal_comp.deriv\n  \u00b7 have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf\n    rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt <| hf',\n      Complex.ofReal_zero]\n\nlemma deriv.ofReal_comp' {f : \u211d \u2192 \u211d} :\n    deriv (fun x : \u211d \u21a6 (f x : \u2102)) = (fun x \u21a6 ((deriv f) x : \u2102)) :=\n  funext fun _ \u21a6 deriv.ofReal_comp\n\nlemma deriv.comp_ofReal {e : \u2102 \u2192 \u2102} {z : \u211d} (hf : DifferentiableAt \u2102 e z) :\n    deriv (fun x : \u211d \u21a6 e x) z = deriv e z :=\n  hf.hasDerivAt.comp_ofReal.deriv\n\nlemma deriv.comp_ofReal' {e : \u2102 \u2192 \u2102} (hf : Differentiable \u2102 e) :\n    deriv (fun x : \u211d \u21a6 e x) = fun (x : \u211d) \u21a6 deriv e x :=\n  funext fun _ \u21a6 deriv.comp_ofReal (hf.differentiableAt)\n\n/-%%\n\\begin{lemma}[PartialIntegration]\\label{PartialIntegration}\\lean{PartialIntegration}\\leanok\nLet $f, g$ be once differentiable functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ so that $fg'$\nand $f'g$ are both integrable, and $f*g (x)\\to 0$ as $x\\to 0^+,\\infty$.\nThen\n$$\n\\int_0^\\infty f(x)g'(x) dx = -\\int_0^\\infty f'(x)g(x)dx.\n$$\n\\end{lemma}\n%%-/\n/-- *Need differentiability, and decay at `0` and `\u221e`* -/\nlemma PartialIntegration (f g : \u211d \u2192 \u2102)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0))\n    (gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0))\n    (lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  simpa using integral_Ioi_mul_deriv_eq_deriv_mul\n    (fun x hx \u21a6 fDiff.hasDerivAt (Ioi_mem_nhds hx))\n    (fun x hx \u21a6 gDiff.hasDerivAt (Ioi_mem_nhds hx))\n    fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n/-%%\n\\begin{proof}\\leanok\nPartial integration.\n\\end{proof}\n%%-/\n\nlemma PartialIntegration_of_support_in_Icc {a b : \u211d} (f g : \u211d \u2192 \u2102) (ha : 0 < a) (h : a \u2264 b)\n    (fSupp : f.support \u2286 Set.Icc a b)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (gDiff : DifferentiableOn \u211d g (Ioi 0))\n    (fderivCont : ContinuousOn (deriv f) (Ioi 0))\n    (gderivCont : ContinuousOn (deriv g) (Ioi 0)) :\n    \u222b x in Ioi 0, f x * deriv g x = -\u222b x in Ioi 0, deriv f x * g x := by\n  have Icc_sub : Icc a b \u2286 Ioi 0 := (Icc_subset_Ioi_iff h).mpr ha\n  have fderivSupp := Function.support_deriv_subset_Icc fSupp\n  have fgSupp : (f * g).support \u2286 Icc a b := Function.support_mul_subset_of_subset fSupp\n  have fDerivgInt : IntegrableOn (f * deriv g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fDiff.continuousOn.mono Icc_sub\n    \u00b7 exact gderivCont.mono Icc_sub\n  have gDerivfInt : IntegrableOn (deriv f * g) (Ioi 0) := by\n    apply (integrableOn_iff_integrable_of_support_subset <|\n           Function.support_mul_subset_of_subset fderivSupp).mp\n    apply ContinuousOn.integrableOn_Icc <| ContinuousOn.mul ?_ ?_\n    \u00b7 exact fderivCont.mono Icc_sub\n    \u00b7 exact gDiff.continuousOn.mono Icc_sub\n  have lim_at_zero : Tendsto (f * g) (\ud835\udcdd[>]0) (\ud835\udcdd 0) := TendstoAtZero_of_support_in_Icc (f * g) ha fgSupp\n  have lim_at_inf : Tendsto (f * g) atTop (\ud835\udcdd 0) := TendstoAtTop_of_support_in_Icc (f * g) fgSupp\n  apply PartialIntegration f g fDiff gDiff fDerivgInt gDerivfInt lim_at_zero lim_at_inf\n\n/-%%\nIn this section, we define the Mellin transform (already in Mathlib, thanks to David Loeffler),\nprove its inversion formula, and\nderive a number of important properties of some special functions and bumpfunctions.\n\nDef: (Already in Mathlib)\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be the function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\n[Note: My preferred way to think about this is that we are integrating over the multiplicative\ngroup $\\mathbb{R}_{>0}$, multiplying by a (not necessarily unitary!) character $|\\cdot|^s$, and\nintegrating with respect to the invariant Haar measure $dx/x$. This is very useful in the kinds\nof calculations carried out below. But may be more difficult to formalize as things now stand. So\nwe might have clunkier calculations, which ``magically'' turn out just right - of course they're\nexplained by the aforementioned structure...]\n\n%%-/\n\n\n/-%%\n\\begin{definition}[MellinTransform]\\label{MellinTransform}\\lean{MellinTransform}\\leanok\nLet $f$ be a function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. We define the Mellin transform of\n$f$ to be\nthe function $\\mathcal{M}(f)$ from $\\mathbb{C}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx.$$\n\\end{definition}\n[Note: already exists in Mathlib, with some good API.]\n%%-/\nnoncomputable def MellinTransform (f : \u211d \u2192 \u2102) (s : \u2102) : \u2102 :=\n  \u222b x in Ioi 0, f x * x ^ (s - 1)\n\nlocal notation (name := mellintransform) \"\ud835\udcdc\" => MellinTransform\n/-%%\n\\begin{definition}[MellinInverseTransform]\\label{MellinInverseTransform}\n\\lean{MellinInverseTransform}\\leanok\nLet $F$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$. We define the Mellin inverse transform of\n$F$ to be the function $\\mathcal{M}^{-1}(F)$ from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\mathcal{M}^{-1}(F)(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}F(s)x^{-s}ds,$$\nwhere $\\sigma$ is sufficiently large (say $\\sigma>2$).\n\\end{definition}\n%%-/\nnoncomputable def MellinInverseTransform (F : \u2102 \u2192 \u2102) (\u03c3 : \u211d) (x : \u211d) : \u2102 :=\n  VerticalIntegral' (fun s \u21a6 x ^ (-s) * F s) \u03c3\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_lt]\\label{PerronInverseMellin_lt}\\lean{PerronInverseMellin_lt}\n\\leanok\nLet $0 < t < x$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function\n$$F: s\\mapsto t^s/s(s+1)$$ is equal to\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds\n= 0.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_lt {t x : \u211d} (tpos : 0 < t) (t_lt_x : t < x) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 0 := by\n  dsimp [MellinInverseTransform, VerticalIntegral']\n  have xpos : 0 < x := lt_trans tpos t_lt_x\n  simp only [one_div, mul_inv_rev, inv_I, neg_mul, neg_eq_zero, mul_eq_zero, I_ne_zero,\n    inv_eq_zero, ofReal_eq_zero, pi_ne_zero, OfNat.ofNat_ne_zero, or_self, false_or]\n  convert Perron.formulaLtOne (div_pos tpos xpos) ((div_lt_one xpos).mpr t_lt_x) \u03c3pos using 2\n  ext1 s\n  convert Perron.f_mul_eq_f tpos xpos s using 1\n  ring\n/-%%\n\\begin{proof}\\leanok\n\\uses{Perron.formulaLtOne}\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[PerronInverseMellin_gt]\\label{PerronInverseMellin_gt}\\lean{PerronInverseMellin_gt}\n\\leanok\nLet $0 < x < t$ and $\\sigma>0$. Then the inverse Mellin transform of the Perron function is equal\nto\n$$\\frac{1}{2\\pi i}\\int_{(\\sigma)}\\frac{t^s}{s(s+1)}x^{-s}ds = 1 - x / t.$$\n\\end{lemma}\n%%-/\nlemma PerronInverseMellin_gt {t x : \u211d} (xpos : 0 < x) (x_lt_t : x < t) {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3) :\n    MellinInverseTransform (Perron.f t) \u03c3 x = 1 - x / t := by\n  dsimp [MellinInverseTransform]\n  have tpos : 0 < t := by linarith\n  have txinv_gtOne : 1 < t / x := (one_lt_div xpos).mpr x_lt_t\n  rw [\u2190 smul_eq_mul]\n  convert Perron.formulaGtOne txinv_gtOne \u03c3pos using 2\n  \u00b7 congr\n    ext1 s\n    convert Perron.f_mul_eq_f tpos xpos s using 1\n    ring\n  \u00b7 field_simp\n/-%%\n\\begin{proof}\n\\uses{Perron.formulaGtOne}\\leanok\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-% ** Wrong delimiters on purpose **\nUnnecessary lemma:\n%\\begin{lemma}[MellinInversion_aux1]\\label{MellinInversion_aux1}\\lean{MellinInversion_aux1}\\leanok\nLet $f$ be differentiable on $(0,\\infty)$, and assume that $f(x)x^s\\to 0$ as $x\\to 0$, and that\n$f(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f(x)x^s\\frac{dx}{x} = \\frac{1}{s}\\int_0^\\infty f'(x)x^{s} dx.\n$$\n%\\end{lemma}\n%-/\nlemma MellinInversion_aux1 {f : \u211d \u2192 \u2102} {s : \u2102} (s_ne_zero : s \u2260 0)\n    (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, f x * x ^ s / x = - \u222b x in Ioi 0, (deriv f x) * x ^ s / s := by\n  sorry\n\n/-% ** Wrong delimiters on purpose **\n\\begin{proof}\n\\uses{PartialIntegration}\nPartial integration.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux2]\\label{MellinInversion_aux2}\\lean{MellinInversion_aux2}\\leanok\nLet $f$ be twice differentiable on $(0,\\infty)$, and assume that $f'(x)x^s\\to 0$ as $x\\to 0$, and\nthat $f'(x)x^s\\to 0$.\nThen\n$$\n\\int_0^\\infty f'(x)x^{s} dx = -\\int_0^\\infty f''(x)x^{s+1}\\frac{1}{(s+1)}dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux2 {f : \u211d \u2192 \u2102} (s : \u2102) (fDiff : DifferentiableOn \u211d f (Ioi 0))\n    (fDiff2 : DifferentiableOn \u211d (deriv f) (Ioi 0))\n    (hfs : Tendsto (fun x \u21a6 deriv f x * x ^ s) (\ud835\udcdd[>]0) (\ud835\udcdd 0))\n    (hfinf : Tendsto (fun x \u21a6 deriv f x * x ^ s) atTop (\ud835\udcdd 0)) :\n    \u222b x in Ioi 0, (deriv f x) * x ^ s =\n      -\u222b x in Ioi 0, (deriv (deriv f) x) * x ^ (s + 1) / (s + 1) := by\n  sorry\n/-%\n\\begin{proof}\n\\uses{PartialIntegration, MellinInversion_aux1}\nPartial integration. (Apply Lemma \\ref{MellinInversion_aux1} to the function $f'$ and $s+1$.)\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux3]%\\label{MellinInversion_aux3}\\lean{MellinInversion_aux3}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen the map  $(x,s) \\mapsto f(x)x^s/(s(s+1))$ is absolutely integrable on\n$(0,\\infty)\\times\\{\\Re s = \\sigma\\}$ for any $\\sigma>0$.\n\\end{lemma}\n%-/\nlemma MellinInversion_aux3 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    IntegrableOn (fun (\u27e8x, t\u27e9 : \u211d \u00d7 \u211d) \u21a6 f x * x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * ((\u03c3 + t * I) + 1)))\n      ((Ioi 0).prod (univ : Set \u211d)) := by\n  sorry\n/-%\n\\begin{proof}\nPut absolute values and estimate.\n\\end{proof}\n%-/\n\n/-% ** Wrong delimiters on purpose **\n\\begin{lemma}[MellinInversion_aux4]%\\label{MellinInversion_aux4}\\lean{MellinInversion_aux4}\\leanok\nGiven $f$ and $\\sigma$, assume that $f(x)x^\\sigma$ is absolutely integrable on $(0,\\infty)$.\nThen we can interchange orders of integration\n$$\n\\int_{(\\sigma)}\\int_0^\\infty f(x)x^{s+1}\\frac{1}{s(s+1)}dx ds =\n\\int_0^\\infty\n\\int_{(\\sigma)}f(x)x^{s+1}\\frac{1}{s(s+1)}ds dx.\n$$\n\\end{lemma}\n%-/\nlemma MellinInversion_aux4 {f : \u211d \u2192 \u2102} (\u03c3 : \u211d) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_negOne : \u03c3 \u2260 -1)\n    (fInt : IntegrableOn (fun x \u21a6 f x * (x : \u2102) ^ (\u03c3 : \u2102)) (Ioi 0)) :\n    VerticalIntegral (fun s \u21a6 \u222b x in Ioi 0, f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 =\n      \u222b x in Ioi 0, VerticalIntegral (fun s \u21a6 f x * (x : \u2102) ^ (s + 1) / (s * (s + 1))) \u03c3 := by\n  sorry -- `MeasureTheory.integral_prod` and `MeasureTheory.integral_swap` should be useful here\n/-%\n\\begin{proof}\n\\uses{MellinInversion_aux3}\nFubini-Tonelli.\n\\end{proof}\n%-/\n\nlemma MellinTransform_eq : \ud835\udcdc = mellin := by unfold mellin MellinTransform; simp_rw [smul_eq_mul, mul_comm]\n\nlemma MellinInverseTransform_eq (\u03c3 : \u211d) (f : \u2102 \u2192 \u2102) :\n    MellinInverseTransform f \u03c3 = mellinInv \u03c3 f := by\n  unfold mellinInv MellinInverseTransform VerticalIntegral' VerticalIntegral\n  beta_reduce; ext x\n  rw [\u2190 smul_assoc, smul_eq_mul (a' := I), div_mul]; simp\n\n/-%%\n\\begin{theorem}[MellinInversion]\\label{MellinInversion}\\lean{MellinInversion}\\leanok\nLet $f$ be a twice differentiable function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$, and\nlet $\\sigma$\nbe sufficiently large. Then\n$$f(x) = \\frac{1}{2\\pi i}\\int_{(\\sigma)}\\mathcal{M}(f)(s)x^{-s}ds.$$\n\\end{theorem}\n\n%[Note: How ``nice''? Schwartz (on $(0,\\infty)$) is certainly enough. As we formalize\n%this, we can add whatever\n% conditions are necessary for the proof to go through.]\n%%-/\ntheorem MellinInversion (\u03c3 : \u211d) {f : \u211d \u2192 \u2102} {x : \u211d} (hx : 0 < x) (hf : MellinConvergent f \u03c3)\n    (hFf : VerticalIntegrable (mellin f) \u03c3) (hfx : ContinuousAt f x) :\n    MellinInverseTransform (\ud835\udcdc f) \u03c3 x = f x := by\n  rw [MellinTransform_eq, MellinInverseTransform_eq, mellin_inversion \u03c3 f hx hf hFf hfx]\n/-%%\n\\begin{proof}\\leanok\n\\uses{PartialIntegration, formulaLtOne, formulaGtOne, MellinTransform,\nMellinInverseTransform, PerronInverseMellin_gt, PerronInverseMellin_lt}\n%MellinInversion_aux1, MellinInversion_aux2, MellinInversion_aux3,\n%MellinInversion_aux4, }\nThe proof is from [Goldfeld-Kontorovich 2012].\nIntegrate by parts twice (assuming $f$ is twice differentiable, and all occurring\nintegrals converge absolutely, and\nboundary terms vanish).\n$$\n\\mathcal{M}(f)(s) = \\int_0^\\infty f(x)x^{s-1}dx = - \\int_0^\\infty f'(x)x^s\\frac{1}{s}dx\n= \\int_0^\\infty f''(x)x^{s+1}\\frac{1}{s(s+1)}dx.\n$$\nWe now have at least quadratic decay in $s$ of the Mellin transform. Inserting this\nformula into the inversion formula and Fubini-Tonelli (we now have absolute\nconvergence!) gives:\n$$\nRHS = \\frac{1}{2\\pi i}\\left(\\int_{(\\sigma)}\\int_0^\\infty\n  f''(t)t^{s+1}\\frac{1}{s(s+1)}dt\\right) x^{-s}ds\n$$\n$$\n= \\int_0^\\infty f''(t) t \\left( \\frac{1}{2\\pi i}\n\\int_{(\\sigma)}(t/x)^s\\frac{1}{s(s+1)}ds\\right) dt.\n$$\nApply the Perron formula to the inside:\n$$\n= \\int_x^\\infty f''(t) t \\left(1-\\frac{x}{t}\\right)dt\n= -\\int_x^\\infty f'(t) dt\n= f(x),\n$$\nwhere we integrated by parts (undoing the first partial integration), and finally\napplied the fundamental theorem of calculus (undoing the second).\n\\end{proof}\n%%-/\n\n\n/-%%\nFinally, we need Mellin Convolutions and properties thereof.\n\\begin{definition}[MellinConvolution]\\label{MellinConvolution}\\lean{MellinConvolution}\n\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$. Then we define the\nMellin convolution of $f$ and $g$ to be the function $f\\ast g$ from $\\mathbb{R}_{>0}$\nto $\\mathbb{C}$ defined by\n$$(f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}.$$\n\\end{definition}\n%%-/\nnoncomputable def MellinConvolution (f g : \u211d \u2192 \ud835\udd42) (x : \u211d) : \ud835\udd42 :=\n  \u222b y in Ioi 0, f y * g (x / y) / y\n\n/-%%\nLet us start with a simple property of the Mellin convolution.\n\\begin{lemma}[MellinConvolutionSymmetric]\\label{MellinConvolutionSymmetric}\n\\lean{MellinConvolutionSymmetric}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{R}$ or $\\mathbb{C}$, for $x\\neq0$,\n$$\n  (f\\ast g)(x)=(g\\ast f)(x)\n  .\n$$\n\\end{lemma}\n%%-/\nlemma MellinConvolutionSymmetric (f g : \u211d \u2192 \ud835\udd42) {x : \u211d} (xpos: 0 < x) :\n    MellinConvolution f g x = MellinConvolution g f x := by\n  unfold MellinConvolution\n  calc\n    _ = \u222b y in Ioi 0, f (y * x) * g (1 / y) / y := ?_\n    _ = _ := ?_\n  \u00b7 rw [\u2190 integral_comp_mul_right_I0i_haar (fun y \u21a6 f y * g (x / y)) xpos]\n    simp [div_mul_cancel_right\u2080 <| ne_of_gt xpos]\n  \u00b7 convert (integral_comp_inv_I0i_haar fun y \u21a6 f (y * x) * g (1 / y)).symm using 3\n    rw [one_div_one_div, mul_comm, mul_comm_div, one_mul]\n/-%%\n\\begin{proof}\\leanok\n  \\uses{MellinConvolution}\n  By Definition \\ref{MellinConvolution},\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(y)g(x/y)\\frac{dy}{y}\n  $$\n  in which we change variables to $z=x/y$:\n  $$\n    (f\\ast g)(x) = \\int_0^\\infty f(x/z)g(z)\\frac{dz}{z}\n    =(g\\ast f)(x)\n    .\n  $$\n\\end{proof}\n%%-/\n\n/-%%\nThe Mellin transform of a convolution is the product of the Mellin transforms.\n\\begin{theorem}[MellinConvolutionTransform]\\label{MellinConvolutionTransform}\n\\lean{MellinConvolutionTransform}\\leanok\nLet $f$ and $g$ be functions from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ such that\n\\begin{equation}\n  (x,y)\\mapsto f(y)\\frac{g(x/y)}yx^{s-1}\n  \\label{eq:assm_integrable_Mconv}\n\\end{equation}\nis absolutely integrable on $[0,\\infty)^2$.\nThen\n$$\\mathcal{M}(f\\ast g)(s) = \\mathcal{M}(f)(s)\\mathcal{M}(g)(s).$$\n\\end{theorem}\n%%-/\nlemma MellinConvolutionTransform (f g : \u211d \u2192 \u2102) (s : \u2102)\n    (hf : IntegrableOn (fun x y \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)).uncurry\n      (Ioi 0 \u00d7\u02e2 Ioi 0)) :\n    \ud835\udcdc (MellinConvolution f g) s = \ud835\udcdc f s * \ud835\udcdc g s := by\n  dsimp [MellinTransform, MellinConvolution]\n  set f\u2081 : \u211d \u00d7 \u211d \u2192 \u2102 := fun \u27e8x, y\u27e9 \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, \u222b (y : \u211d) in Ioi 0, f\u2081 (x, y) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f\u2081 (x, y) := set_integral_integral_swap _ hf\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x / y) / \u2191y * \u2191x ^ (s - 1) := rfl\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * g (x * y / y) / \u2191y * \u2191(x * y) ^ (s - 1) * y := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, \u222b (x : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * (g x * \u2191x ^ (s - 1)) := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, f y * \u2191y ^ (s - 1) * \u222b (x : \u211d) in Ioi 0, g x * \u2191x ^ (s - 1) := ?_\n    _ = _ := integral_mul_right _ _\n  <;> try (rw [set_integral_congr (by simp)]; intro y hy; simp only [ofReal_mul])\n  \u00b7 simp only [integral_mul_right]; rfl\n  \u00b7 simp only [integral_mul_right]\n    have := integral_comp_mul_right_Ioi (fun x \u21a6 f y * g (x / y) / (y : \u2102) * (x : \u2102) ^ (s - 1)) 0 hy\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := slitPlane_ne_zero (Or.inl hy)\n    field_simp at this \u22a2\n    rw [this]\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro x hx\n    have y_ne_zero\u211d : y \u2260 0 := ne_of_gt (mem_Ioi.mp hy)\n    have y_ne_zero\u2102 : (y : \u2102) \u2260 0 := by exact_mod_cast y_ne_zero\u211d\n    field_simp [mul_cpow_ofReal_nonneg (LT.lt.le hx) (LT.lt.le hy)]\n    ring\n  \u00b7 apply integral_mul_left\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform,MellinConvolution}\nBy Definitions \\ref{MellinTransform} and \\ref{MellinConvolution}\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}\\frac{dy}ydx\n$$\nBy (\\ref{eq:assm_integrable_Mconv}) and Fubini's theorem,\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(x/y)x^{s-1}dx\\frac{dy}y\n$$\nin which we change variables from $x$ to $z=x/y$:\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\int_0^\\infty \\int_0^\\infty f(y)g(z)y^{s-1}z^{s-1}dzdy\n$$\nwhich, by Definition \\ref{MellinTransform}, is\n$$\n  \\mathcal M(f\\ast g)(s)=\n  \\mathcal M(f)(s)\\mathcal M(g)(s)\n  .\n$$\n\n\\end{proof}\n%%-/\n\n/-%%\nLet $\\psi$ be a bumpfunction.\n\\begin{theorem}[SmoothExistence]\\label{SmoothExistence}\\lean{SmoothExistence}\\leanok\nThere exists a smooth (once differentiable would be enough), nonnegative ``bumpfunction'' $\\psi$,\n supported in $[1/2,2]$ with total mass one:\n$$\n\\int_0^\\infty \\psi(x)\\frac{dx}{x} = 1.\n$$\n\\end{theorem}\n%%-/\n\nattribute [- simp] one_div in\n\nlemma SmoothExistence : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n    \u03a8.support \u2286 Icc (1 / 2) 2 \u2227 \u222b x in Ici 0, \u03a8 x / x = 1 := by\n  suffices h : \u2203 (\u03a8 : \u211d \u2192 \u211d), (ContDiff \u211d \u22a4 \u03a8) \u2227 (\u2200 x, 0 \u2264 \u03a8 x) \u2227\n      \u03a8.support \u2286 Set.Icc (1 / 2) 2 \u2227 0 < \u222b x in Set.Ici 0, \u03a8 x / x by\n    rcases h with \u27e8\u03a8, h\u03a8, h\u03a8nonneg, h\u03a8supp, h\u03a8pos\u27e9\n    let c := (\u222b x in Ici 0, \u03a8 x / x)\n    use fun y \u21a6 \u03a8 y / c\n    refine \u27e8h\u03a8.div_const c, fun y \u21a6 div_nonneg (h\u03a8nonneg y) (le_of_lt h\u03a8pos), ?_, ?_\u27e9\n    \u00b7 rw [Function.support_div, Function.support_const (ne_of_lt h\u03a8pos).symm, inter_univ]\n      convert h\u03a8supp\n    \u00b7 simp only [div_right_comm _ c _, integral_div c, div_self <| ne_of_gt h\u03a8pos]\n\n  have := smooth_urysohn_support_Ioo (a := 1 / 2) (b := 1) (c := 3/2) (d := 2) (by linarith)\n    (by linarith)\n  rcases this with \u27e8\u03a8, h\u03a8ContDiff, _, h\u03a80, h\u03a81, h\u03a8Support\u27e9\n  use \u03a8, h\u03a8ContDiff\n  unfold indicator at h\u03a80 h\u03a81\n  simp only [mem_Icc, Pi.one_apply, Pi.le_def, mem_Ioo] at h\u03a80 h\u03a81\n  simp only [h\u03a8Support, subset_def, mem_Ioo, mem_Icc, and_imp]\n  split_ands\n  \u00b7 exact fun x \u21a6 le_trans (by simp [apply_ite]) (h\u03a80 x)\n  \u00b7 exact fun y hy hy' \u21a6 \u27e8by linarith, by linarith\u27e9\n  \u00b7 rw [integral_pos_iff_support_of_nonneg]\n    \u00b7 simp only [Function.support_div, measurableSet_Ici, Measure.restrict_apply', h\u03a8Support, Function.support_id]\n      have : (Ioo (1 / 2 : \u211d) 2 \u2229 (Iio 0 \u222a Ioi 0) \u2229 Ici 0) = Ioo (1 / 2) 2 := by\n        ext x\n        simp only [mem_inter_iff, mem_Ioo, mem_Ici, mem_Iio, mem_Ioi,\n          mem_union, not_lt, and_true, not_le]\n        constructor\n        \u00b7 exact fun h \u21a6 h.left.left\n        \u00b7 intro h\n          simp only [h, and_self, lt_or_lt_iff_ne, ne_eq, true_and]\n          constructor <;> linarith [h.left]\n      simp only [this, volume_Ioo, ENNReal.ofReal_pos, sub_pos, gt_iff_lt]\n      linarith\n    \u00b7 simp_rw [Pi.le_def, Pi.zero_apply]\n      intro y\n      by_cases h : y \u2208 Function.support \u03a8\n      . apply div_nonneg <| le_trans (by simp [apply_ite]) (h\u03a80 y)\n        rw [h\u03a8Support, mem_Ioo] at h\n        linarith [h.left]\n      . simp only [Function.mem_support, ne_eq, not_not] at h\n        simp [h]\n    \u00b7 have : (fun x \u21a6 \u03a8 x / x).support \u2286 Icc (1 / 2) 2 := by\n        rw [Function.support_div, h\u03a8Support]\n        apply subset_trans (by apply inter_subset_left) Ioo_subset_Icc_self\n      apply (integrableOn_iff_integrable_of_support_subset this).mp\n      apply ContinuousOn.integrableOn_compact isCompact_Icc\n      apply ContinuousOn.div h\u03a8ContDiff.continuous.continuousOn continuousOn_id ?_\n      simp only [mem_Icc, ne_eq, and_imp, id_eq]\n      intros\n      linarith\n/-%%\n\\begin{proof}\\leanok\n\\uses{smooth-ury}\nSame idea as Urysohn-type argument.\n\\end{proof}\n%%-/\n\nlemma mem_within_strip (\u03c3\u2081 \u03c3\u2082 : \u211d) :\n  {s : \u2102 | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} \u2208 \ud835\udcdf {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082} := by simp\n\nlemma MellinOfPsi_aux {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {s : \u2102} (hs : s \u2260 0) :\n    \u222b (x : \u211d) in Ioi 0, (\u03a8 x) * (x : \u2102) ^ (s - 1) =\n    - (1 / s) * \u222b (x : \u211d) in Ioi 0, (deriv \u03a8 x) * (x : \u2102) ^ s := by\n  let g {s : \u2102} := fun (x : \u211d)  \u21a6 x ^ s / s\n  have gderiv {s : \u2102} (hs : s \u2260 0) {x: \u211d} (hx : x \u2208 Ioi 0) :\n      deriv g x = x ^ (s - 1) := by\n    have := HasDerivAt.cpow_const (c := s) (hasDerivAt_id (x : \u2102)) (Or.inl hx)\n    simp_rw [mul_one, id_eq] at this\n    rw [deriv_div_const, deriv.comp_ofReal (e := fun x \u21a6 x ^ s)]\n    \u00b7 rw [this.deriv, mul_div_right_comm, div_self hs, one_mul]\n    \u00b7 apply hasDerivAt_deriv_iff.mp\n      simp only [this.deriv, this]\n  calc\n    _ =  \u222b (x : \u211d) in Ioi 0, \u2191(\u03a8 x) * deriv (@g s) x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv (fun x \u21a6 \u2191(\u03a8 x)) x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * @g s x := ?_\n    _ = -\u222b (x : \u211d) in Ioi 0, deriv \u03a8 x * x ^ s / s := by simp only [mul_div, g]\n    _ = _ := ?_\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro _ hx\n    simp only [gderiv hs hx]\n  \u00b7 apply PartialIntegration_of_support_in_Icc (\u03a8 \u00b7) g\n      (a := 1 / 2) (b := 2) (by norm_num) (by norm_num)\n    \u00b7 simpa only [Function.support_subset_iff, ne_eq, ofReal_eq_zero]\n    \u00b7 exact (Differentiable.ofReal_comp_iff.mpr (diff\u03a8.differentiable (by norm_num))).differentiableOn\n    \u00b7 refine DifferentiableOn.div_const ?_ s\n      intro a ha\n      refine DifferentiableAt.differentiableWithinAt ?_\n      apply DifferentiableAt.comp_ofReal (e := fun x \u21a6 x ^ s)\n      apply DifferentiableAt.cpow differentiableAt_id' <| differentiableAt_const s\n      exact Or.inl ha\n    \u00b7 simp only [deriv.ofReal_comp']\n      apply Continuous.continuousOn\n      apply Continuous.comp (g := ofReal') continuous_ofReal <| diff\u03a8.continuous_deriv (by norm_num)\n    \u00b7 apply ContinuousOn.congr (f := fun (x : \u211d) \u21a6 (x : \u2102) ^ (s - 1)) ?_ fun x hx \u21a6 gderiv hs hx\n      refine ContinuousOn.cpow ?_ continuousOn_const (by simp)\n      exact Continuous.continuousOn (by continuity)\n  \u00b7 congr; funext; congr\n    apply (hasDerivAt_deriv_iff.mpr ?_).ofReal_comp.deriv\n    exact diff\u03a8.contDiffAt.differentiableAt (by norm_num)\n  \u00b7 simp only [neg_mul, neg_inj]\n    conv => lhs; rhs; intro; rw [\u2190 mul_one_div, mul_comm]\n    rw [integral_mul_left]\n\n/-%%\nThe $\\psi$ function has Mellin transform $\\mathcal{M}(\\psi)(s)$ which is entire and decays (at\nleast) like $1/|s|$.\n\\begin{theorem}[MellinOfPsi]\\label{MellinOfPsi}\\lean{MellinOfPsi}\\leanok\nThe Mellin transform of $\\psi$ is\n$$\\mathcal{M}(\\psi)(s) =  O\\left(\\frac{1}{|s|}\\right),$$\nas $|s|\\to\\infty$ with $\\sigma_1 \\le \\Re(s) \\le \\sigma_2$.\n\\end{theorem}\n\n[Of course it decays faster than any power of $|s|$, but it turns out that we will just need one\npower.]\n%%-/\nlemma MellinOfPsi {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {\u03c3\u2081 \u03c3\u2082 : \u211d} (\u03c3\u2081pos : 0 < \u03c3\u2081) :\n    (fun s \u21a6 \u2016\ud835\udcdc (\u03a8 \u00b7) s\u2016)\n    =O[Filter.principal {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082}]\n      fun s \u21a6 1 / \u2016s\u2016 := by\n  let f := fun (x : \u211d) \u21a6 \u2016deriv \u03a8 x\u2016\n  have cont : ContinuousOn f (Icc (1 / 2) 2) :=\n    (Continuous.comp (by continuity) <| diff\u03a8.continuous_deriv (by norm_num)).continuousOn\n  obtain \u27e8a, _, max\u27e9 := isCompact_Icc.exists_isMaxOn (f := f) (by norm_num) cont\n  rw [Asymptotics.isBigO_iff]\n  use f a * 2 ^ \u03c3\u2082 * (3 / 2)\n  filter_upwards [mem_within_strip \u03c3\u2081 \u03c3\u2082] with s hs\n  have s_ne_zero: s \u2260 0 := fun h \u21a6 by linarith [zero_re \u25b8 h \u25b8 hs.1]\n  simp only [MellinTransform, f, MellinOfPsi_aux diff\u03a8 supp\u03a8 s_ne_zero, norm_norm, norm_mul]\n  conv => rhs; rw [mul_comm]\n  gcongr; simp\n  calc\n    _ \u2264 \u222b (x : \u211d) in Ioi 0, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016 := ?_\n    _ = \u222b (x : \u211d) in Icc (1 / 2) 2, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016 := ?_\n    _ \u2264 \u2016\u222b (x : \u211d) in Icc (1 / 2) 2, \u2016(deriv \u03a8 x * (x : \u2102) ^ s)\u2016\u2016 := le_abs_self _\n    _ \u2264 _ := ?_\n  \u00b7 simp_rw [norm_integral_le_integral_norm]\n  \u00b7 apply SetIntegral.integral_eq_integral_inter_of_support_subset_Icc\n    \u00b7 simp only [Function.support_abs, Function.support_mul, Function.support_ofReal]\n      apply subset_trans (by apply inter_subset_left) <| Function.support_deriv_subset_Icc supp\u03a8\n    \u00b7 exact (Icc_subset_Ioi_iff (by norm_num)).mpr (by norm_num)\n  \u00b7 have := intervalIntegral.norm_integral_le_of_norm_le_const' (C := f a * 2 ^ \u03c3\u2082)\n      (f := fun x \u21a6 f x * \u2016(x : \u2102) ^ s\u2016) (a := (1 / 2 : \u211d)) ( b := 2) (by norm_num) ?_\n    \u00b7 simp only [Real.norm_eq_abs, Complex.norm_eq_abs, abs_ofReal, map_mul] at this \u22a2\n      rwa [(by norm_num: |(2 : \u211d) - 1 / 2| = 3 / 2),\n          intervalIntegral.integral_of_le (by norm_num), \u2190 integral_Icc_eq_integral_Ioc] at this\n    \u00b7 intro x hx;\n      have f_bound := isMaxOn_iff.mp max x hx\n      have pow_bound : \u2016(x : \u2102) ^ s\u2016 \u2264 2 ^ \u03c3\u2082 := by\n        rw [Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos (by linarith [mem_Icc.mp hx])]\n        have xpos : 0 \u2264 x := by linarith [(mem_Icc.mp hx).1]\n        have h := rpow_le_rpow xpos (mem_Icc.mp hx).2 (by linarith : 0 \u2264 s.re)\n        exact le_trans h <| rpow_le_rpow_of_exponent_le (by norm_num) hs.2\n      convert mul_le_mul f_bound pow_bound (norm_nonneg _) ?_ using 1 <;> simp [f]\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform, SmoothExistence}\nIntegrate by parts:\n$$\n\\left|\\int_0^\\infty \\psi(x)x^s\\frac{dx}{x}\\right| =\n\\left|-\\int_0^\\infty \\psi'(x)\\frac{x^{s}}{s}dx\\right|\n$$\n$$\n\\le \\frac{1}{|s|} \\int_{1/2}^2|\\psi'(x)|x^{\\Re(s)}dx.\n$$\nSince $\\Re(s)$ is bounded, the right-hand side is bounded by a\nconstant times $1/|s|$.\n\\end{proof}\n%%-/\n\n/-%%\nWe can make a delta spike out of this bumpfunction, as follows.\n\\begin{definition}[DeltaSpike]\\label{DeltaSpike}\\lean{DeltaSpike}\\leanok\n\\uses{SmoothExistence}\nLet $\\psi$ be a bumpfunction supported in $[1/2,2]$. Then for any $\\epsilon>0$, we define the\ndelta spike $\\psi_\\epsilon$ to be the function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$\\psi_\\epsilon(x) = \\frac{1}{\\epsilon}\\psi\\left(x^{\\frac{1}{\\epsilon}}\\right).$$\n\\end{definition}\n%%-/\n\nnoncomputable def DeltaSpike (\u03a8 : \u211d \u2192 \u211d) (\u03b5 : \u211d) : \u211d \u2192 \u211d :=\n  fun x \u21a6 \u03a8 (x ^ (1 / \u03b5)) / \u03b5\n\n/-%%\nThis spike still has mass one:\n\\begin{lemma}[DeltaSpikeMass]\\label{DeltaSpikeMass}\\lean{DeltaSpikeMass}\\leanok\nFor any $\\epsilon>0$, we have\n$$\\int_0^\\infty \\psi_\\epsilon(x)\\frac{dx}{x} = 1.$$\n\\end{lemma}\n%%-/\n\nlemma DeltaSpikeMass {\u03a8 : \u211d \u2192 \u211d} (mass_one: \u222b x in Ioi 0, \u03a8 x / x = 1) {\u03b5 : \u211d}\n    (\u03b5pos : 0 < \u03b5) : \u222b x in Ioi 0, ((DeltaSpike \u03a8 \u03b5) x) / x = 1 :=\n  calc\n    _ = \u222b (x : \u211d) in Ioi 0, (|1/\u03b5| * x ^ (1 / \u03b5 - 1)) \u2022\n      ((fun z \u21a6 (\u03a8 z) / z) (x ^ (1 / \u03b5))) := by\n      apply set_integral_congr_ae measurableSet_Ioi\n      filter_upwards with x hx\n      simp only [mem_Ioi, smul_eq_mul, abs_of_pos (one_div_pos.mpr \u03b5pos)]\n      symm; calc\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / x ^ (1 / \u03b5)) * x ^ (1 / \u03b5 - 1) * (1 / \u03b5) := by ring\n        _ = _ := by rw [rpow_sub hx, rpow_one]\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / x ^ (1 / \u03b5) * x ^ (1 / \u03b5) / x) * (1/ \u03b5) := by ring\n        _ = _ := by rw [div_mul_cancel\u2080 _ (ne_of_gt (rpow_pos_of_pos hx (1/\u03b5)))]\n        _ = (\u03a8 (x ^ (1 / \u03b5)) / \u03b5 / x) := by ring\n    _ = 1 := by\n      rw [integral_comp_rpow_Ioi (fun z \u21a6 (\u03a8 z) / z), \u2190 mass_one]\n      simp only [ne_eq, div_eq_zero_iff, one_ne_zero, \u03b5pos.ne', or_self, not_false_eq_true]\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{DeltaSpike}\nSubstitute $y=x^{1/\\epsilon}$, and use the fact that $\\psi$ has mass one, and that $dx/x$ is Haar\nmeasure.\n\\end{proof}\n%%-/\n\nlemma DeltaSpikeSupport_aux {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2):\n    (fun x \u21a6 if x < 0 then 0 else DeltaSpike \u03a8 \u03b5 x).support \u2286 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) := by\n  unfold DeltaSpike\n  simp only [one_div, Function.support_subset_iff, ne_eq, ite_eq_left_iff, not_lt, div_eq_zero_iff,\n    not_forall, exists_prop, mem_Icc, and_imp]\n  intro x hx h; push_neg at h\n  have := supp\u03a8 <| Function.mem_support.mpr h.1\n  simp only [one_div, mem_Icc] at this\n  have hl := (le_rpow_inv_iff_of_pos (by norm_num) hx \u03b5pos).mp this.1\n  rw [inv_rpow (by norm_num) \u03b5, \u2190 rpow_neg (by norm_num)] at hl\n  refine \u27e8hl, (rpow_inv_le_iff_of_pos ?_ (by norm_num) \u03b5pos).mp this.2\u27e9\n  linarith [(by apply rpow_nonneg (by norm_num) : 0 \u2264 (2 : \u211d) ^ (-\u03b5))]\n\nlemma DeltaSpikeSupport' {\u03a8 : \u211d \u2192 \u211d} {\u03b5 x : \u211d} (\u03b5pos : 0 < \u03b5) (xnonneg : 0 \u2264 x)\n    (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2) :\n    DeltaSpike \u03a8 \u03b5 x \u2260 0 \u2192 x \u2208 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) := by\n  intro h\n  have : (fun x \u21a6 if x < 0 then 0 else DeltaSpike \u03a8 \u03b5 x) x = DeltaSpike \u03a8 \u03b5 x := by simp [xnonneg]\n  rw [\u2190 this] at h\n  exact (Function.support_subset_iff.mp <| DeltaSpikeSupport_aux \u03b5pos supp\u03a8) _ h\n\nlemma DeltaSpikeSupport {\u03a8 : \u211d \u2192 \u211d} {\u03b5 x : \u211d} (\u03b5pos : 0 < \u03b5) (xnonneg : 0 \u2264 x)\n    (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2) :\n    x \u2209 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) \u2192 DeltaSpike \u03a8 \u03b5 x = 0 := by\n  contrapose!; exact DeltaSpikeSupport' \u03b5pos xnonneg supp\u03a8\n\nlemma DeltaSpikeContinuous {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (diff\u03a8 : ContDiff \u211d 1 \u03a8) :\n    Continuous (fun x \u21a6 DeltaSpike \u03a8 \u03b5 x) := by\n  apply (Continuous.comp (g := \u03a8) diff\u03a8.continuous _).div_const\n  exact Continuous.rpow_const continuous_id fun _ \u21a6 Or.inr <| div_nonneg (by norm_num) \u03b5pos.le\n\nlemma DeltaSpikeOfRealContinuous {\u03a8 : \u211d \u2192 \u211d} {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) (diff\u03a8 : ContDiff \u211d 1 \u03a8) :\n    Continuous (fun x \u21a6 (DeltaSpike \u03a8 \u03b5 x : \u2102)) :=\n  Continuous.comp continuous_ofReal <| DeltaSpikeContinuous \u03b5pos diff\u03a8\n\n/-%%\nThe Mellin transform of the delta spike is easy to compute.\n\\begin{theorem}[MellinOfDeltaSpike]\\label{MellinOfDeltaSpike}\\lean{MellinOfDeltaSpike}\\leanok\nFor any $\\epsilon>0$, the Mellin transform of $\\psi_\\epsilon$ is\n$$\\mathcal{M}(\\psi_\\epsilon)(s) = \\mathcal{M}(\\psi)\\left(\\epsilon s\\right).$$\n\\end{theorem}\n%%-/\ntheorem MellinOfDeltaSpike (\u03a8 : \u211d \u2192 \u211d) {\u03b5 : \u211d} (\u03b5pos : \u03b5 > 0) (s : \u2102) :\n    \ud835\udcdc ((DeltaSpike \u03a8 \u03b5) \u00b7) s = \ud835\udcdc (\u03a8 \u00b7) (\u03b5 * s) := by\n  unfold MellinTransform DeltaSpike\n  rw [\u2190 integral_comp_rpow_Ioi (fun z \u21a6 ((\u03a8 z) : \u2102) * (z : \u2102) ^ ((\u03b5 : \u2102) * s - 1))\n    (one_div_ne_zero (ne_of_gt \u03b5pos))]\n  apply set_integral_congr_ae measurableSet_Ioi\n  filter_upwards with x hx\n\n  -- Simple algebra, would be nice if some tactic could handle this\n  have log_x_real: (Complex.log (x : \u2102)).im = 0 := by\n    rw [\u2190 ofReal_log, ofReal_im]\n    exact LT.lt.le hx\n  rw [div_eq_mul_inv, ofReal_mul, abs_of_pos (one_div_pos.mpr \u03b5pos)]\n  simp only [real_smul, ofReal_mul, ofReal_div, ofReal_one, Complex.ofReal_rpow hx]\n  rw [\u2190 Complex.cpow_mul _ ?_ ?_, mul_sub]\n  \u00b7 simp only [\u2190 mul_assoc, ofReal_sub, ofReal_div, ofReal_one, mul_one, ofReal_inv]\n    symm\n    \u00b7 rw [one_div_mul_cancel (by exact slitPlane_ne_zero (Or.inl \u03b5pos)), mul_comm (1 / (\u03b5:\u2102)),\n          mul_comm, \u2190 mul_assoc, \u2190 mul_assoc]\n      rw [\u2190 Complex.cpow_add _ _ (by exact slitPlane_ne_zero (Or.inl hx))]; ring_nf\n  \u00b7 simp [im_mul_ofReal, log_x_real, zero_mul, pi_pos]\n  \u00b7 simp [im_mul_ofReal, log_x_real, zero_mul, pi_nonneg]\n/-%%\n\\begin{proof}\\leanok\n\\uses{DeltaSpike, MellinTransform}\nSubstitute $y=x^{1/\\epsilon}$, use Haar measure; direct calculation.\n\\end{proof}\n%%-/\n\n/-%%\nIn particular, for $s=1$, we have that the Mellin transform of $\\psi_\\epsilon$ is $1+O(\\epsilon)$.\n\\begin{corollary}[MellinOfDeltaSpikeAt1]\\label{MellinOfDeltaSpikeAt1}\\lean{MellinOfDeltaSpikeAt1}\n\\leanok\nFor any $\\epsilon>0$, we have\n$$\\mathcal{M}(\\psi_\\epsilon)(1) =\n\\mathcal{M}(\\psi)(\\epsilon).$$\n\\end{corollary}\n%%-/\n\nlemma MellinOfDeltaSpikeAt1 (\u03a8 : \u211d \u2192 \u211d) {\u03b5 : \u211d} (\u03b5pos : \u03b5 > 0) :\n    \ud835\udcdc ((DeltaSpike \u03a8 \u03b5) \u00b7) 1 = \ud835\udcdc (\u03a8 \u00b7) \u03b5 := by\n  convert MellinOfDeltaSpike \u03a8 \u03b5pos 1; simp [mul_one]\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinOfDeltaSpike, DeltaSpikeMass}\nThis is immediate from the above theorem.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[MellinOfDeltaSpikeAt1_asymp]\\label{MellinOfDeltaSpikeAt1_asymp}\n\\lean{MellinOfDeltaSpikeAt1_asymp}\\leanok\nAs $\\epsilon\\to 0$, we have\n$$\\mathcal{M}(\\psi_\\epsilon)(1) = 1+O(\\epsilon).$$\n\\end{lemma}\n%%-/\nlemma MellinOfDeltaSpikeAt1_asymp {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    (mass_one : \u222b x in Set.Ioi 0, \u03a8 x / x = 1) :\n    (fun (\u03b5 : \u211d) \u21a6 (\ud835\udcdc (\u03a8 \u00b7) \u03b5) - 1) =O[\ud835\udcdd[>]0] id := by\n  have diff : DifferentiableWithinAt \u211d (fun (\u03b5 : \u211d) \u21a6 \ud835\udcdc (\u03a8 \u00b7) \u03b5 - 1) (Ioi 0) 0 := by\n    apply DifferentiableAt.differentiableWithinAt\n    simp only [differentiableAt_sub_const_iff, MellinTransform_eq]\n    refine DifferentiableAt.comp_ofReal ?_\n    refine mellin_differentiableAt_of_isBigO_rpow (a := 1) (b := -1) ?_ ?_ (by simp) ?_ (by simp)\n    \u00b7 apply ContinuousOn.locallyIntegrableOn (Continuous.continuousOn ?_) (by simp)\n      have := diff\u03a8.continuous; continuity\n    \u00b7 apply Asymptotics.IsBigO.trans_le (g' := fun _ \u21a6 (0 : \u211d)) ?_ (by simp)\n      apply BigO_zero_atTop_of_support_in_Icc (a := 1 / 2) (b := 2)\n      rwa [\u03a8.support_ofReal]\n    \u00b7 apply Asymptotics.IsBigO.trans_le (g' := fun _ \u21a6 (0 : \u211d)) ?_ (by simp)\n      apply BigO_zero_atZero_of_support_in_Icc (a := 1 / 2) (b := 2) (ha := (by norm_num))\n      rwa [\u03a8.support_ofReal]\n  have := ofReal_zero \u25b8 diff.isBigO_sub\n  simp only [sub_sub_sub_cancel_right, sub_zero] at this\n  convert this\n  simp only [MellinTransform, zero_sub, sub_right_inj, cpow_neg_one, \u2190 div_eq_mul_inv, \u2190 ofReal_div]\n  rw [\u2190 ofReal_one, \u2190 mass_one]; convert integral_ofReal.symm\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform,MellinOfDeltaSpikeAt1,SmoothExistence}\nBy Lemma \\ref{MellinOfDeltaSpikeAt1},\n$$\n  \\mathcal M(\\psi_\\epsilon)(1)=\\mathcal M(\\psi)(\\epsilon)\n$$\nwhich by Definition \\ref{MellinTransform} is\n$$\n  \\mathcal M(\\psi)(\\epsilon)=\\int_0^\\infty\\psi(x)x^{\\epsilon-1}dx\n  .\n$$\nSince $\\psi(x) x^{\\epsilon-1}$ is integrable (because $\\psi$ is continuous and compactly supported),\n$$\n  \\mathcal M(\\psi)(\\epsilon)-\\int_0^\\infty\\psi(x)\\frac{dx}x=\\int_0^\\infty\\psi(x)(x^{\\epsilon-1}-x^{-1})dx\n  .\n$$\nBy Taylor's theorem,\n$$\n  x^{\\epsilon-1}-x^{-1}=O(\\epsilon)\n$$\nso, since $\\psi$ is absolutely integrable,\n$$\n  \\mathcal M(\\psi)(\\epsilon)-\\int_0^\\infty\\psi(x)\\frac{dx}x=O(\\epsilon)\n  .\n$$\nWe conclude the proof using Theorem \\ref{SmoothExistence}.\n\\end{proof}\n%%-/\n\n/-%%\nLet $1_{(0,1]}$ be the function from $\\mathbb{R}_{>0}$ to $\\mathbb{C}$ defined by\n$$1_{(0,1]}(x) = \\begin{cases}\n1 & \\text{ if }x\\leq 1\\\\\n0 & \\text{ if }x>1\n\\end{cases}.$$\nThis has Mellin transform\n\\begin{theorem}[MellinOf1]\\label{MellinOf1}\\lean{MellinOf1}\\leanok\nThe Mellin transform of $1_{(0,1]}$ is\n$$\\mathcal{M}(1_{(0,1]})(s) = \\frac{1}{s}.$$\n\\end{theorem}\n[Note: this already exists in mathlib]\n%%-/\nlemma MellinOf1 (s : \u2102) (h : s.re > 0) : \ud835\udcdc ((fun x \u21a6 if 0 < x \u2227 x \u2264 1 then 1 else 0)) s = 1 / s := by\n  convert (hasMellin_one_Ioc h).right using 1\n  apply set_integral_congr_ae measurableSet_Ioi\n  filter_upwards with _ _; rw [smul_eq_mul, mul_comm]; congr\n\n/-%%\n\\begin{proof}\\leanok\n\\uses{MellinTransform}\nThis is a straightforward calculation.\n\\end{proof}\n%%-/\n\n/-%%\nWhat will be essential for us is properties of the smooth version of $1_{(0,1]}$, obtained as the\n Mellin convolution of $1_{(0,1]}$ with $\\psi_\\epsilon$.\n\\begin{definition}[Smooth1]\\label{Smooth1}\\lean{Smooth1}\n\\uses{MellinOf1, MellinConvolution}\\leanok\nLet $\\epsilon>0$. Then we define the smooth function $\\widetilde{1_{\\epsilon}}$ from\n$\\mathbb{R}_{>0}$ to $\\mathbb{C}$ by\n$$\\widetilde{1_{\\epsilon}} = 1_{(0,1]}\\ast\\psi_\\epsilon.$$\n\\end{definition}\n%%-/\nnoncomputable def Smooth1 (\u03a8 : \u211d \u2192 \u211d) (\u03b5 : \u211d) : \u211d \u2192 \u211d :=\n  MellinConvolution (fun x \u21a6 if 0 < x \u2227 x \u2264 1 then 1 else 0) (DeltaSpike \u03a8 \u03b5)\n\n/-%%\n\\begin{lemma}[Smooth1Properties_estimate]\\label{Smooth1Properties_estimate}\n\\lean{Smooth1Properties_estimate}\\leanok\nFor $\\epsilon>0$,\n$$\n  \\log2>\\frac{1-2^{-\\epsilon}}\\epsilon\n$$\n\\end{lemma}\n%%-/\n\nlemma Smooth1Properties_estimate {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n    (1 - 2 ^ (-\u03b5)) / \u03b5 < Real.log 2 := by\n  apply (div_lt_iff' \u03b5pos).mpr\n  have : 1 - 1 / (2 : \u211d) ^ \u03b5 = ((2 : \u211d) ^ \u03b5 - 1) / (2 : \u211d) ^ \u03b5 := by\n    rw [sub_div, div_self (by positivity)]\n  rw [\u2190 Real.log_rpow (by norm_num), rpow_neg (by norm_num), inv_eq_one_div (2 ^ \u03b5), this]\n  set c := (2 : \u211d) ^ \u03b5\n  have hc : 1 < c := by\n    rw [\u2190 rpow_zero (2 : \u211d)]\n    apply Real.rpow_lt_rpow_of_exponent_lt (by norm_num) \u03b5pos\n  apply (div_lt_iff' (by positivity)).mpr <| lt_sub_iff_add_lt'.mp ?_\n  let f := (fun x \u21a6 x * Real.log x - x)\n  rw [(by simp [f] : -1 = f 1), (by simp : c * Real.log c - c = f c)]\n  have mono: StrictMonoOn f <| Ici 1 := by\n    refine strictMonoOn_of_deriv_pos (convex_Ici _) ?_ ?_\n    \u00b7 apply ContinuousOn.sub (ContinuousOn.mul continuousOn_id ?_) continuousOn_id\n      apply ContinuousOn.log continuousOn_id\n      intro x hx; simp only [mem_Ici] at hx; simp only [id_eq, ne_eq]; linarith\n    \u00b7 intro x hx; simp only [nonempty_Iio, interior_Ici', mem_Ioi] at hx\n      funext; dsimp only [f]\n      rw [deriv_sub, deriv_mul, deriv_log, deriv_id'', one_mul, mul_inv_cancel]; simp\n      \u00b7 exact log_pos hx\n      \u00b7 linarith\n      \u00b7 simp only [differentiableAt_id']\n      \u00b7 simp only [differentiableAt_log_iff, ne_eq]; linarith\n      \u00b7 apply DifferentiableAt.mul differentiableAt_id' <| DifferentiableAt.log differentiableAt_id' ?_\n        linarith\n      \u00b7 simp only [differentiableAt_id']\n  exact mono (by rw [mem_Ici]) (mem_Ici.mpr <| le_of_lt hc) hc\n/-%%\n\\begin{proof}\\leanok\nLet $c:=2^\\epsilon > 1$, in terms of which we wish to prove\n$$\n  -1 < c \\log c - c .\n$$\nLetting $f(x):=x\\log x - x$, we can rewrite this as $f(1) < f(c)$.\nSince\n$$\n  \\frac {d}{dx}f(x) = \\log x > 0 ,\n$$\n$f$ is monotone increasing on [1, \\infty), and we are done.\n\\end{proof}\n%%-/\n\n\n/-%%\nIn particular, we have the following two properties.\n\\begin{lemma}[Smooth1Properties_below]\\label{Smooth1Properties_below}\n\\lean{Smooth1Properties_below}\\leanok\nFix $\\epsilon>0$. There is an absolute constant $c>0$ so that:\nIf $0 < x \\leq (1-c\\epsilon)$, then\n$$\\widetilde{1_{\\epsilon}}(x) = 1.$$\n\\end{lemma}\n%%-/\n\nlemma Smooth1Properties_below_aux {x \u03b5 : \u211d} (hx : x \u2264 1 - (2:\u211d).log * \u03b5) (\u03b5pos: 0 < \u03b5) :\n    x < 2 ^ (-\u03b5) := by\n  calc\n    x \u2264 1 - (2 : \u211d).log * \u03b5 := hx\n    _ < 2 ^ (-\u03b5) := ?_\n  rw [sub_lt_iff_lt_add, add_comm, \u2190 sub_lt_iff_lt_add]\n  exact (div_lt_iff \u03b5pos).mp <| Smooth1Properties_estimate \u03b5pos\n\nlemma Smooth1Properties_below {\u03a8 : \u211d \u2192 \u211d} (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2)\n    (\u03b5 : \u211d) (\u03b5pos: 0 < \u03b5) (mass_one : \u222b x in Ioi 0, \u03a8 x / x = 1) :\n    \u2203 (c : \u211d), 0 < c \u2227 \u2200 (x : \u211d), 0 < x \u2192 x \u2264 1 - c * \u03b5 \u2192 Smooth1 \u03a8 \u03b5 x = 1 := by\n  set c := (2 : \u211d).log; use c\n  constructor; exact log_pos (by norm_num)\n  intro x xpos hx\n  have hx2 := Smooth1Properties_below_aux hx \u03b5pos\n  rewrite [\u2190 DeltaSpikeMass mass_one \u03b5pos]\n  unfold Smooth1 MellinConvolution\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, indicator (Ioc 0 1) (fun y \u21a6 DeltaSpike \u03a8 \u03b5 (x / y) / \u2191y) y := ?_\n    _ = \u222b (y : \u211d) in Ioi 0, DeltaSpike \u03a8 \u03b5 (x / y) / y := ?_\n    _ = _ := integral_comp_div_I0i_haar (fun y \u21a6 DeltaSpike \u03a8 \u03b5 y) xpos\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro y hy\n    by_cases h : y \u2264 1 <;> simp [indicator, mem_Ioi.mp hy, h]\n  \u00b7 rw [set_integral_congr (by simp)]\n    intro y hy\n    simp only [indicator_apply_eq_self, mem_Ioc, not_and, not_le, div_eq_zero_iff]\n    intro hy2; replace hy2 := hy2 <| mem_Ioi.mp hy\n    by_cases h : y = 0; right; exact h; left\n    apply DeltaSpikeSupport \u03b5pos ?_ supp\u03a8\n    \u00b7 simp only [mem_Icc, not_and, not_le]; intro\n      linarith [(by apply (div_lt_iff (by linarith)).mpr; nlinarith : x / y < 2 ^ (-\u03b5))]\n    \u00b7 rw [le_div_iff (by linarith), zero_mul]; exact xpos.le\n/-%%\n\\begin{proof}\\leanok\n\\uses{Smooth1, MellinConvolution,DeltaSpikeMass, Smooth1Properties_estimate}\nOpening the definition, we have that the Mellin convolution of $1_{(0,1]}$ with $\\psi_\\epsilon$ is\n$$\n\\int_0^\\infty 1_{(0,1]}(y)\\psi_\\epsilon(x/y)\\frac{dy}{y}\n=\n\\int_0^1 \\psi_\\epsilon(x/y)\\frac{dy}{y}.\n$$\nThe support of $\\psi_\\epsilon$ is contained in $[1/2^\\epsilon,2^\\epsilon]$, so it suffices to consider\n$y \\in [1/2^\\epsilon x,2^\\epsilon x]$ for nonzero contributions. If $x < 2^{-\\epsilon}$, then the integral is the same as that over $(0,\\infty)$:\n$$\n\\int_0^1 \\psi_\\epsilon(x/y)\\frac{dy}{y}\n=\n\\int_0^\\infty \\psi_\\epsilon(x/y)\\frac{dy}{y},\n$$\nin which we change variables to $z=x/y$ (using $x>0$):\n$$\n\\int_0^\\infty \\psi_\\epsilon(x/y)\\frac{dy}{y}\n=\n\\int_0^\\infty \\psi_\\epsilon(z)\\frac{dz}{z},\n$$\nwhich is equal to one by Lemma \\ref{DeltaSpikeMass}.\nWe then choose\n$$\n  c:=\\log 2,\n$$\nwhich satisfies\n$$\n  c > \\frac{1-2^{-\\epsilon}}\\epsilon\n$$\nby Lemma \\ref{Smooth1Properties_estimate}, so\n$$\n  1-c\\epsilon < 2^{-\\epsilon}.\n$$\n\\end{proof}\n%%-/\n\nlemma Smooth1Properties_above_aux {x \u03b5 : \u211d} (hx : 1 + (2 * (2:\u211d).log) * \u03b5 \u2264 x) (h\u03b5 : \u03b5 \u2208 Ioo 0 1) :\n    2 ^ \u03b5 < x := by\n  calc\n    x \u2265 1 + (2 * (2 : \u211d).log) * \u03b5 := hx\n    _ > 2 ^ \u03b5 := ?_\n  refine lt_add_of_sub_left_lt <| (div_lt_iff h\u03b5.1).mp ?_\n  calc\n    2 * (2 : \u211d).log > 2 * (1 - 2 ^ (-\u03b5)) / \u03b5 := ?_\n    _ > 2 ^ \u03b5 * (1 - 2 ^ (-\u03b5)) / \u03b5 := ?_\n    _ = (2 ^ \u03b5 - 1) / \u03b5 := ?_\n  \u00b7 have := (mul_lt_mul_left (a := 2) (by norm_num)).mpr <| Smooth1Properties_estimate h\u03b5.1\n    field_simp at this; simp [this]\n  \u00b7 have : (2 : \u211d) ^ \u03b5 < 2 := by\n      nth_rewrite 1 [\u2190 pow_one 2]\n      convert rpow_lt_rpow_of_exponent_lt (x := 2) (by norm_num) h\u03b5.2 <;> norm_num\n    have pos: 0 < (1 - 2 ^ (-\u03b5)) / \u03b5 := by\n      refine div_pos ?_ h\u03b5.1\n      rw [sub_pos, \u2190 pow_zero 2]\n      convert rpow_lt_rpow_of_exponent_lt (x := 2) (by norm_num) (neg_lt_zero.mpr h\u03b5.1); norm_num\n    have := (mul_lt_mul_right pos).mpr this\n    ring_nf at this \u22a2\n    exact this\n  \u00b7 have : (2 : \u211d) ^ \u03b5 * (2 : \u211d) ^ (-\u03b5) = (2 : \u211d) ^ (\u03b5 - \u03b5) := by\n      rw [\u2190 rpow_add (by norm_num), add_neg_self, sub_self]\n    conv => lhs; lhs; ring_nf; rhs; simp [this]\n\nlemma Smooth1Properties_above_aux2 {x y \u03b5 : \u211d} (h\u03b5 : \u03b5 \u2208 Ioo 0 1) (hy : y \u2208 Ioc 0 1)\n  (hx2 : 2 ^ \u03b5 < x) :\n    2 < (x / y) ^ (1 / \u03b5) := by\n  obtain \u27e8\u03b5pos, \u03b51\u27e9 := h\u03b5\n  obtain \u27e8ypos, y1\u27e9 := hy\n  calc\n    _ > (2 ^ \u03b5 / y) ^ (1 / \u03b5) := ?_\n    _ = 2 / y ^ (1 / \u03b5) := ?_\n    _ \u2265 2 / y := ?_\n    _ \u2265 2 := ?_\n  \u00b7 rw [gt_iff_lt, div_rpow, div_rpow, lt_div_iff, mul_comm_div, div_self, mul_one]\n    <;> try positivity\n    \u00b7 exact rpow_lt_rpow (by positivity) hx2 (by positivity)\n    \u00b7 exact LT.lt.le <| lt_trans (by positivity) hx2\n  \u00b7 rw [div_rpow, \u2190 rpow_mul, mul_div_cancel\u2080 1 <| ne_of_gt \u03b5pos, rpow_one] <;> positivity\n  \u00b7 have : y ^ (1 / \u03b5) \u2264 y := by\n      nth_rewrite 2 [\u2190 rpow_one y]\n      exact rpow_le_rpow_of_exponent_ge ypos y1 (by linarith [one_lt_one_div \u03b5pos \u03b51])\n    have pos : 0 < y ^ (1 / \u03b5) := by apply rpow_pos_of_pos <| ypos\n    rw [ge_iff_le, div_le_iff, div_mul_eq_mul_div, le_div_iff', mul_comm] <;> try linarith\n  \u00b7 rw [ge_iff_le, le_div_iff <| ypos]; exact (mul_le_iff_le_one_right zero_lt_two).mpr y1\n/-%%\n\\begin{lemma}[Smooth1Properties_above]\\label{Smooth1Properties_above}\n\\lean{Smooth1Properties_above}\\leanok\nFix $0<\\epsilon<1$. There is an absolute constant $c>0$ so that:\nif $x\\geq (1+c\\epsilon)$, then\n$$\\widetilde{1_{\\epsilon}}(x) = 0.$$\n\\end{lemma}\n%%-/\nlemma Smooth1Properties_above {\u03a8 : \u211d \u2192 \u211d} (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2)\n    {\u03b5 : \u211d} (h\u03b5 : \u03b5 \u2208 Ioo 0 1) :\n    \u2203 (c : \u211d), 0 < c \u2227 \u2200 (x : \u211d), 1 + c * \u03b5 \u2264 x \u2192 Smooth1 \u03a8 \u03b5 x = 0 := by\n  set c := 2 * Real.log 2; use c\n  constructor\n  \u00b7 simp only [c, zero_lt_two, mul_pos_iff_of_pos_left]; exact log_pos (by norm_num)\n  intro x hx\n  have hx2 := Smooth1Properties_above_aux hx h\u03b5\n  unfold Smooth1 MellinConvolution\n  simp only [ite_mul, one_mul, zero_mul, RCLike.ofReal_real_eq_id, id_eq]\n  apply set_integral_eq_zero_of_forall_eq_zero\n  intro y hy\n  have ypos := mem_Ioi.mp hy\n  by_cases y1 : y \u2264 1; swap; simp [ypos, y1]\n  simp only [mem_Ioi.mp hy, y1, and_self, \u2193reduceIte, div_eq_zero_iff]; left\n  apply DeltaSpikeSupport h\u03b5.1 ?_ supp\u03a8\n  simp only [mem_Icc, not_and, not_le]\n  swap; suffices h : 2 ^ \u03b5 < x / y by\n    linarith [(by apply rpow_pos_of_pos (by norm_num) : 0 < (2 : \u211d) ^ \u03b5)]\n  all_goals\n  try intro\n  have : x / y = ((x / y) ^ (1 / \u03b5)) ^ \u03b5 := by\n    rw [\u2190 rpow_mul]\n    simp only [one_div, inv_mul_cancel (ne_of_gt h\u03b5.1), rpow_one]\n    apply div_nonneg_iff.mpr; left;\n    exact \u27e8(le_trans (rpow_pos_of_pos (by norm_num) \u03b5).le) hx2.le, ypos.le\u27e9\n  rw [this]\n  refine rpow_lt_rpow (by norm_num) ?_ h\u03b5.1\n  exact Smooth1Properties_above_aux2 h\u03b5 \u27e8ypos, y1\u27e9 hx2\n/-%%\n\\begin{proof}\\leanok\n\\uses{Smooth1, MellinConvolution, Smooth1Properties_estimate}\nAgain the Mellin convolution is\n$$\\int_0^1 \\psi_\\epsilon(x/y)\\frac{dy}{y},$$\nbut now if $x > 2^\\epsilon$, then the support of $\\psi_\\epsilon$ is disjoint\nfrom the region of integration, and hence the integral is zero.\nWe choose\n$$\n  c:=2\\log 2\n  .\n$$\nBy Lemma \\ref{Smooth1Properties_estimate},\n$$\n  c > 2\\frac{1-2^{-\\epsilon}}\\epsilon > 2^\\epsilon\\frac{1-2^{-\\epsilon}}\\epsilon\n  =\n  \\frac{2^\\epsilon-1}\\epsilon,\n$$\nso\n$$\n  1+c\\epsilon > 2^\\epsilon.\n$$\n\\end{proof}\n%%-/\n\nlemma DeltaSpikeNonNeg_of_NonNeg {\u03a8 : \u211d \u2192 \u211d} (\u03a8nonneg : \u2200 x > 0, 0 \u2264 \u03a8 x)\n     {x \u03b5 : \u211d} (xpos : 0 < x) (\u03b5pos : 0 < \u03b5) :\n    0 \u2264 DeltaSpike \u03a8 \u03b5 x := by\n  dsimp [DeltaSpike]\n  have : 0 < x ^ (1 / \u03b5) := by positivity\n  have : 0 \u2264 \u03a8 (x ^ (1 / \u03b5)) := \u03a8nonneg _ this\n  positivity\n\nlemma MellinConvNonNeg_of_NonNeg {f g : \u211d \u2192 \u211d} (f_nonneg : \u2200 x > 0, 0 \u2264 f x)\n    (g_nonneg : \u2200 x > 0, 0 \u2264 g x) {x : \u211d} (xpos : 0 < x) :\n    0 \u2264 MellinConvolution f g x := by\n  dsimp [MellinConvolution]\n  apply MeasureTheory.set_integral_nonneg\n  \u00b7 exact measurableSet_Ioi\n  \u00b7 intro y ypos; simp only [mem_Ioi] at ypos\n    have : 0 \u2264 f y := f_nonneg _ ypos\n    have : 0 < x / y := by positivity\n    have : 0 \u2264 g (x / y) := g_nonneg _ this\n    positivity\n\n/-%%\n\\begin{lemma}[Smooth1Nonneg]\\label{Smooth1Nonneg}\\lean{Smooth1Nonneg}\\leanok\nIf $\\psi$ is nonnegative, then $\\widetilde{1_{\\epsilon}}(x)$ is nonnegative.\n\\end{lemma}\n%%-/\nlemma Smooth1Nonneg {\u03a8 : \u211d \u2192 \u211d} (\u03a8nonneg : \u2200 x > 0, 0 \u2264 \u03a8 x) {\u03b5 x : \u211d} (xpos : 0 < x)\n    (\u03b5pos : 0 < \u03b5) : 0 \u2264 Smooth1 \u03a8 \u03b5 x := by\n  dsimp [Smooth1]\n  apply MellinConvNonNeg_of_NonNeg ?_ ?_ xpos\n  \u00b7 intro y hy; by_cases h : y \u2264 1 <;> simp [h, hy]\n  \u00b7 intro y ypos; exact DeltaSpikeNonNeg_of_NonNeg \u03a8nonneg ypos \u03b5pos\n/-%%\n\\begin{proof}\\uses{Smooth1, MellinConvolution, DeltaSpike}\\leanok\nBy Definitions \\ref{Smooth1}, \\ref{MellinConvolution} and \\ref{DeltaSpike}\n$$\n  \\widetilde{1_\\epsilon}(x)=\\int_0^\\infty 1_{(0,1]}(y)\\frac1\\epsilon\\psi((x/y)^{\\frac1\\epsilon}) \\frac{dy}y\n$$\nand all the factors in the integrand are nonnegative.\n\\end{proof}\n%%-/\n\nlemma Smooth1LeOne_aux {x \u03b5 : \u211d} {\u03a8 : \u211d \u2192 \u211d} (xpos : 0 < x) (\u03b5pos : 0 < \u03b5)\n    (mass_one : \u222b x in Ioi 0, \u03a8 x / x = 1) :\n    \u222b (y : \u211d) in Ioi 0, \u03a8 ((x / y) ^ (1 / \u03b5)) / \u03b5 / y = 1 := by\n    calc\n      _ = \u222b (y : \u211d) in Ioi 0, (\u03a8 (y ^ (1 / \u03b5)) / \u03b5) / y := ?_\n      _ = \u222b (y : \u211d) in Ioi 0, \u03a8 y / y := ?_\n      _ = 1 := mass_one\n    \u00b7 have := integral_comp_div_I0i_haar (fun y \u21a6 \u03a8 ((x / y) ^ (1 / \u03b5)) / \u03b5) xpos\n      convert this.symm using 1\n      congr; funext y; congr; field_simp [mul_comm]\n    \u00b7 have := integral_comp_rpow_I0i_haar_real (fun y \u21a6 \u03a8 y) (one_div_ne_zero \u03b5pos.ne')\n      field_simp [ \u2190 this, abs_of_pos <| one_div_pos.mpr \u03b5pos]\n\n/-%%\n\\begin{lemma}[Smooth1LeOne]\\label{Smooth1LeOne}\\lean{Smooth1LeOne}\\leanok\nIf $\\psi$ is nonnegative and has mass one, then $\\widetilde{1_{\\epsilon}}(x)\\le 1$, $\\forall x>0$.\n\\end{lemma}\n%%-/\nlemma Smooth1LeOne {\u03a8 : \u211d \u2192 \u211d} (\u03a8nonneg : \u2200 x > 0, 0 \u2264 \u03a8 x)\n    (mass_one : \u222b x in Ioi 0, \u03a8 x / x = 1) {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) :\n    \u2200 (x : \u211d), 0 < x \u2192 Smooth1 \u03a8 \u03b5 x \u2264 1 := by\n  unfold Smooth1 MellinConvolution DeltaSpike\n  intro x xpos\n  have := Smooth1LeOne_aux xpos \u03b5pos mass_one\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, (fun y \u21a6 if y \u2208 Ioc 0 1 then 1 else 0) y * (\u03a8 ((x / y) ^ (1 / \u03b5)) / \u03b5 / y) := ?_\n    _ \u2264 \u222b (y : \u211d) in Ioi 0, (\u03a8 ((x / y) ^ (1 / \u03b5)) / \u03b5) / y := ?_\n    _ = 1 := this\n  \u00b7 rw [set_integral_congr (by simp)]\n    simp only [ite_mul, one_mul, zero_mul, RCLike.ofReal_real_eq_id, id_eq, mem_Ioc]\n    intro y hy; aesop\n  \u00b7 refine set_integral_mono_on ?_ (integrable_of_integral_eq_one this) (by simp) ?_\n    \u00b7 refine Integrable.bdd_mul (integrable_of_integral_eq_one this) ?_ (by use 1; aesop)\n      have : (fun x \u21a6 if 0 < x \u2227 x \u2264 1 then 1 else 0) = indicator (Ioc 0 1) (1 : \u211d \u2192 \u211d) := by\n        aesop\n      simp only [mem_Ioc, this, measurableSet_Ioc, aestronglyMeasurable_indicator_iff]\n      exact aestronglyMeasurable_one\n    \u00b7 simp only [ite_mul, one_mul, zero_mul, RCLike.ofReal_real_eq_id, id_eq]\n      intro y hy\n      by_cases h : y \u2264 1; aesop\n      field_simp [mem_Ioc, h, and_false, reduceIte]\n      apply mul_nonneg\n      \u00b7 apply \u03a8nonneg; exact rpow_pos_of_pos (div_pos xpos <| mem_Ioi.mp hy) _\n      \u00b7 apply inv_nonneg.mpr <| mul_nonneg \u03b5pos.le (mem_Ioi.mp hy).le\n/-%%\n\\begin{proof}\\uses{Smooth1,MellinConvolution,DeltaSpike,SmoothExistence}\\leanok\nBy Definitions \\ref{Smooth1}, \\ref{MellinConvolution} and \\ref{DeltaSpike}\n$$\n  \\widetilde{1_\\epsilon}(x)=\\int_0^\\infty 1_{(0,1]}(y)\\frac1\\epsilon\\psi((x/y)^{\\frac1\\epsilon}) \\frac{dy}y\n$$\nand since $1_{(0,1]}(y)\\le 1$, and all the factors in the integrand are nonnegative,\n$$\n  \\widetilde{1_\\epsilon}(x)\\le\\int_0^\\infty \\frac1\\epsilon\\psi((x/y)^{\\frac1\\epsilon}) \\frac{dy}y\n$$\n(because in mathlib the integral of a non-integrable function is $0$, for the inequality above to be true, we must prove that $\\psi((x/y)^{\\frac1\\epsilon})/y$ is integrable; this follows from the computation below).\nWe then change variables to $z=(x/y)^{\\frac1\\epsilon}$:\n$$\n  \\widetilde{1_\\epsilon}(x)\\le\\int_0^\\infty \\psi(z) \\frac{dz}z\n$$\nwhich by Theorem \\ref{SmoothExistence} is 1.\n\\end{proof}\n%%-/\n\n/-%%\nCombining the above, we have the following three Main Lemmata of this section on the Mellin\ntransform of $\\widetilde{1_{\\epsilon}}$.\n\\begin{lemma}[MellinOfSmooth1a]\\label{MellinOfSmooth1a}\\lean{MellinOfSmooth1a}\\leanok\nFix  $\\epsilon>0$. Then the Mellin transform of $\\widetilde{1_{\\epsilon}}$ is\n$$\\mathcal{M}(\\widetilde{1_{\\epsilon}})(s) =\n\\frac{1}{s}\\left(\\mathcal{M}(\\psi)\\left(\\epsilon s\\right)\\right).$$\n\\end{lemma}\n%%-/\nlemma MellinOfSmooth1a (\u03a8 : \u211d \u2192 \u211d) (diff\u03a8 : ContDiff \u211d 1 \u03a8) (supp\u03a8 : \u03a8.support \u2286 Icc (1 / 2) 2)\n    {\u03b5 : \u211d} (\u03b5pos : 0 < \u03b5) {s : \u2102} (hs : 0 < s.re) : \ud835\udcdc ((Smooth1 \u03a8 \u03b5) \u00b7) s = 1 / s * \ud835\udcdc (\u03a8 \u00b7) (\u03b5 * s) := by\n  let f' : \u211d \u2192 \u2102 := fun x \u21a6 DeltaSpike \u03a8 \u03b5 x\n  let f : \u211d \u2192 \u2102 := fun x \u21a6 DeltaSpike \u03a8 \u03b5 x / x\n  let g : \u211d \u2192 \u2102 := fun x \u21a6 if 0 < x \u2227 x \u2264 1 then 1 else 0\n  let F : \u211d \u00d7 \u211d \u2192 \u2102 := Function.uncurry fun x y \u21a6 f y * g (x / y) * (x : \u2102) ^ (s - 1)\n  let S := {\u27e8x, y\u27e9 : \u211d \u00d7 \u211d | 0 < x  \u2227 x \u2264 y \u2227 2 ^ (-\u03b5) \u2264 y \u2227 y \u2264 2 ^ \u03b5}\n  let F' : \u211d \u00d7 \u211d \u2192 \u2102 := piecewise S (fun \u27e8x, y\u27e9 \u21a6 f y * (x : \u2102) ^ (s - 1))\n     (fun _ \u21a6 0)\n  let Tx := Ioc 0 ((2 : \u211d) ^ \u03b5)\n  let Ty := Icc ((2 : \u211d) ^ (-\u03b5)) ((2 : \u211d) ^ \u03b5)\n\n  have Seq : S = (Tx \u00d7\u02e2 Ty) \u2229 {(x, y) : \u211d \u00d7 \u211d | x \u2264 y} := by\n    ext \u27e8x, y\u27e9; constructor\n    \u00b7 exact fun h \u21a6 \u27e8\u27e8\u27e8h.1, le_trans h.2.1 h.2.2.2\u27e9, \u27e8h.2.2.1, h.2.2.2\u27e9\u27e9, h.2.1\u27e9\n    \u00b7 exact fun h \u21a6  \u27e8h.1.1.1, \u27e8h.2, h.1.2.1, h.1.2.2\u27e9\u27e9\n  have SsubI : S \u2286 Ioi 0 \u00d7\u02e2 Ioi 0 :=\n    fun z hz \u21a6 \u27e8hz.1, lt_of_lt_of_le (by apply rpow_pos_of_pos; norm_num) hz.2.2.1\u27e9\n  have SsubT: S \u2286 Tx \u00d7\u02e2 Ty := by simp_rw [Seq, inter_subset_left]\n  have Smeas : MeasurableSet S := by\n    rw [Seq]; apply MeasurableSet.inter ?_ <| measurableSet_le measurable_fst measurable_snd\n    simp [measurableSet_prod, Tx, Ty]\n\n  have int_F: IntegrableOn F (Ioi 0 \u00d7\u02e2 Ioi 0) := by\n    apply IntegrableOn.congr_fun (f := F') ?_ ?_ (by simp [measurableSet_prod]); swap\n    \u00b7 simp only [F, F', f, g, mul_ite, mul_one, mul_zero, Function.uncurry_apply_pair]\n      intro \u27e8x, y\u27e9 hz\n      by_cases hS : \u27e8x, y\u27e9 \u2208 S <;> simp only [hS, piecewise, hz]\n      <;> simp only [mem_prod, mem_Ioi, mem_setOf_eq, not_and, not_le, S] at hz hS\n      \u00b7 simp [hS, div_pos hz.1 hz.2, (div_le_one hz.2).mpr hS.2.1]\n      \u00b7 by_cases hxy : x / y \u2264 1; swap; simp [hxy]\n        have hy : y \u2209 Icc (2 ^ (-\u03b5)) (2 ^ \u03b5) := by\n          simp only [mem_Icc, not_and, not_le]; exact hS hz.1 <| (div_le_one hz.2).mp hxy\n        simp [DeltaSpikeSupport \u03b5pos hz.2.le supp\u03a8 hy]\n    \u00b7 apply Integrable.piecewise Smeas ?_ integrableOn_zero\n      simp only [IntegrableOn, Measure.restrict_restrict_of_subset SsubI]\n      apply MeasureTheory.Integrable.mono_measure ?_\n      apply MeasureTheory.Measure.restrict_mono' (HasSubset.Subset.eventuallyLE SsubT) le_rfl\n      have : volume.restrict (Tx \u00d7\u02e2 Ty) = (volume.restrict Tx).prod (volume.restrict Ty) := by\n        rw [Measure.prod_restrict, MeasureTheory.Measure.volume_eq_prod]\n      conv => rw [this]; lhs; intro; rw [mul_comm]\n      apply MeasureTheory.Integrable.prod_mul (f := fun x \u21a6 (x : \u2102) ^ (s - 1))\n        (\u03bc := Measure.restrict volume Tx)\n      \u00b7 apply integrableOn_Ioc_iff_integrableOn_Ioo.mpr ?_\n        apply (intervalIntegral.integrableOn_Ioo_cpow_iff (s := s - 1) (t := (2 : \u211d) ^ \u03b5) ?_).mpr\n        \u00b7 simp [hs]\n        \u00b7 apply rpow_pos_of_pos (by norm_num)\n      \u00b7 apply ContinuousOn.integrableOn_compact isCompact_Icc (ContinuousOn.div ?_ ?_ ?_)\n        \u00b7 exact (DeltaSpikeOfRealContinuous \u03b5pos diff\u03a8).continuousOn\n        \u00b7 exact continuous_ofReal.continuousOn\n        \u00b7 intro x hx; simp only [mem_Icc] at hx; simp only [ofReal_ne_zero]\n          linarith [(by apply rpow_pos_of_pos (by norm_num) : (0 : \u211d) < 2 ^ (-\u03b5))]\n\n  have : \ud835\udcdc (MellinConvolution g f') s = \ud835\udcdc g s * \ud835\udcdc f' s := by\n    rw [mul_comm, \u2190 MellinConvolutionTransform f' g s (by convert int_F using 1; field_simp [F, f, f'])]\n    dsimp [MellinTransform]; rw [set_integral_congr (by simp)]\n    intro x hx; simp_rw [MellinConvolutionSymmetric _ _ <| mem_Ioi.mp hx]\n\n  convert this using 1\n  \u00b7 congr; funext x; convert integral_ofReal.symm\n    simp only [MellinConvolution, RCLike.ofReal_div, ite_mul, one_mul, zero_mul, @apply_ite \u211d \u2102,\n      algebraMap.coe_zero, f, g]; rfl\n  \u00b7 rw [MellinOf1 s hs, MellinOfDeltaSpike \u03a8 \u03b5pos s]\n/-%%\n\\begin{proof}\\uses{Smooth1,MellinConvolutionTransform, MellinOfDeltaSpike, MellinOf1, MellinConvolutionSymmetric}\\leanok\nBy Definition \\ref{Smooth1},\n$$\n  \\mathcal M(\\widetilde{1_\\epsilon})(s)\n  =\\mathcal M(1_{(0,1]}\\ast\\psi_\\epsilon)(s)\n  .\n$$\nWe wish to apply Theorem \\ref{MellinConvolutionTransform}.\nTo do so, we must prove that\n$$\n  (x,y)\\mapsto 1_{(0,1]}(y)\\psi_\\epsilon(x/y)/y\n$$\nis integrable on $[0,\\infty)^2$.\nIt is actually easier to do this for the convolution: $\\psi_\\epsilon\\ast 1_{(0,1]}$, so we use Lemma \\ref{MellinConvolutionSymmetric}: for $x\\neq0$,\n$$\n  1_{(0,1]}\\ast\\psi_\\epsilon(x)=\\psi_\\epsilon\\ast 1_{(0,1]}(x)\n  .\n$$\nNow, for $x=0$, both sides of the equation are 0, so the equation also holds for $x=0$.\nTherefore,\n$$\n  \\mathcal M(\\widetilde{1_\\epsilon})(s)\n  =\\mathcal M(\\psi_\\epsilon\\ast 1_{(0,1]})(s)\n  .\n$$\nNow,\n$$\n  (x,y)\\mapsto \\psi_\\epsilon(y)1_{(0,1]}(x/y)\\frac{x^{s-1}}y\n$$\nhas compact support that is bounded away from $y=0$ (specifically $y\\in[2^{-\\epsilon},2^\\epsilon]$ and $x\\in(0,y]$), so it is integrable.\nWe can thus apply Theorem \\ref{MellinConvolutionTransform} and find\n$$\n  \\mathcal M(\\widetilde{1_\\epsilon})(s)\n  =\\mathcal M(\\psi_\\epsilon)(s)\\mathcal M(1_{(0,1]})(s)\n  .\n$$\nBy Lemmas \\ref{MellinOf1} and \\ref{MellinOfDeltaSpike},\n$$\n  \\mathcal M(\\widetilde{1_\\epsilon})(s)\n  =\\frac1s\\mathcal M(\\psi)(\\epsilon s)\n  .\n$$\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[MellinOfSmooth1b]\\label{MellinOfSmooth1b}\\lean{MellinOfSmooth1b}\\leanok\nGiven $0<\\sigma_1\\le\\sigma_2$, for any $s$ such that $\\sigma_1\\le\\mathcal Re(s)\\le\\sigma_2$, we have\n$$\\mathcal{M}(\\widetilde{1_{\\epsilon}})(s) = O\\left(\\frac{1}{\\epsilon|s|^2}\\right).$$\n\\end{lemma}\n%%-/\n", "theoremStatement": "lemma MellinOfSmooth1b {\u03a8 : \u211d \u2192 \u211d} (diff\u03a8 : ContDiff \u211d 1 \u03a8)\n    (supp\u03a8 : \u03a8.support \u2286 Set.Icc (1 / 2) 2)\n    {\u03c3\u2081 \u03c3\u2082 : \u211d} (\u03c3\u2081pos : 0 < \u03c3\u2081) (\u03b5 : \u211d) (\u03b5pos : 0 < \u03b5) :\n    (fun (s : \u2102) \u21a6 \u2016\ud835\udcdc ((Smooth1 \u03a8 \u03b5) \u00b7) s\u2016)\n      =O[Filter.principal {s | \u03c3\u2081 \u2264 s.re \u2227 s.re \u2264 \u03c3\u2082}]\n      fun s \u21a6 1 / (\u03b5 * \u2016s\u2016 ^ 2) ", "theoremName": "MellinOfSmooth1b", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "7ee2c9a981ca450991343fc0e5cc6cba343ac420", "date": "2024-02-26"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/MellinCalculus.lean", "module": "PrimeNumberTheoremAnd.MellinCalculus", "jsonFile": "PrimeNumberTheoremAnd.MellinCalculus.jsonl", "positionMetadata": {"lineInFile": 1585, "tokenPositionInFile": 67000, "theoremPositionInFile": 82}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 260, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", 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"Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have := MellinOfPsi diff\u03a8 supp\u03a8 (mul_pos \u03b5pos \u03c3\u2081pos) (\u03c3\u2082 := \u03b5 * \u03c3\u2082)\n  rw [Asymptotics.isBigO_iff] at this \u22a2\n  obtain \u27e8c, hc\u27e9 := this\n  use c\n  simp only [norm_norm, norm_div, norm_one, eventually_principal, mem_setOf_eq] at hc \u22a2\n  intro s hs\n  rw [MellinOfSmooth1a \u03a8 diff\u03a8 supp\u03a8 \u03b5pos <| gt_of_ge_of_gt hs.1 \u03c3\u2081pos]\n  have : \u2016\ud835\udcdc (fun x \u21a6 \u2191(\u03a8 x)) (\u03b5 * s)\u2016 \u2264 c * (1 / \u2016\u03b5 * s\u2016) := by\n    refine hc (\u03b5 * s) ?_\n    simp only [mul_re, ofReal_re, ofReal_im, zero_mul, sub_zero]\n    exact \u27e8(mul_le_mul_left \u03b5pos).mpr hs.1, (mul_le_mul_left \u03b5pos).mpr hs.2\u27e9\n  convert mul_le_mul_of_nonneg_left (a := 1 / \u2016s\u2016) this ?_ using 1\n  \u00b7 simp\n  \u00b7 simp only [Complex.norm_eq_abs, norm_mul, Real.norm_eq_abs, norm_pow, Complex.abs_abs, one_div,\n    mul_inv_rev, abs_ofReal]; ring_nf\n  \u00b7 exact div_nonneg (by norm_num) (norm_nonneg s)", "proofType": "tactic", "proofLengthLines": 16, "proofLengthTokens": 817}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n\n! This file was ported from Lean 3 source module selberg\n-/\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic\n\n/-!\n# The Selberg Sieve\n\nThis file proves `selberg_bound_simple`, the main theorem of the Selberg.\n-/\n\nnoncomputable section\n\nopen scoped BigOperators Classical Sieve\n\nopen Finset Real Nat Sieve.UpperBoundSieve ArithmeticFunction Sieve\n\nstructure SelbergSieve extends Sieve where mk ::\n  level : \u211d\n  one_le_level : 1 \u2264 level\n\nnamespace SelbergSieve\nset_option quotPrecheck false\n\nvariable (s : SelbergSieve)\nlocal notation3 \"\u03bd\" => Sieve.nu (toSieve s)\nlocal notation3 \"P\" => Sieve.prodPrimes (toSieve s)\nlocal notation3 \"a\" => Sieve.weights (toSieve s)\nlocal notation3 \"X\" => Sieve.totalMass (toSieve s)\nlocal notation3 \"R\" => Sieve.rem (toSieve s)  -- this one seems broken\nlocal notation3 \"g\" => Sieve.selbergTerms (toSieve s)\nlocal notation3 \"y\" => SelbergSieve.level s\nlocal notation3 \"hy\" => SelbergSieve.one_le_level s\n\n--set_option profiler true\n@[simp]\ndef selbergBoundingSum : \u211d :=\n  \u2211 l in divisors P, if l ^ 2 \u2264 y then g l else 0\n\nset_option quotPrecheck false\nlocal notation3 \"S\" => SelbergSieve.selbergBoundingSum s\n\ntheorem selbergBoundingSum_pos :\n    0 < S := by\n  dsimp only [selbergBoundingSum]\n  rw [\u2190 sum_filter]\n  apply sum_pos;\n  \u00b7 intro l hl\n    rw [mem_filter, mem_divisors] at hl\n    \u00b7 apply s.selbergTerms_pos _ (hl.1.1)\n  \u00b7 simp_rw [Finset.Nonempty, mem_filter]; use 1\n    constructor\n    \u00b7 apply one_mem_divisors.mpr s.prodPrimes_ne_zero\n    rw [cast_one, one_pow]\n    exact s.one_le_level\n\ntheorem selbergBoundingSum_ne_zero : S \u2260 0 := by\n  apply _root_.ne_of_gt\n  exact s.selbergBoundingSum_pos\n\ntheorem selbergBoundingSum_nonneg : 0 \u2264 S := _root_.le_of_lt s.selbergBoundingSum_pos\n\ndef selbergWeights : \u2115 \u2192 \u211d := fun d =>\n  if d \u2223 P then\n    (\u03bd d)\u207b\u00b9 * g d * \u03bc d * S\u207b\u00b9 *\n      \u2211 m in divisors P, if (d * m) ^ 2 \u2264 y \u2227 m.Coprime d then g m else 0\n  else 0\n\n-- This notation traditionally uses \u03bb, which is unavailable in lean\nset_option quotPrecheck false\nlocal notation3 \"\u03b3\" => SelbergSieve.selbergWeights s\n\n", "theoremStatement": "theorem selbergWeights_eq_zero_of_not_dvd {d : \u2115} (hd : \u00ac d \u2223 P) :\n    \u03b3 d = 0 ", "theoremName": "SelbergSieve.selbergWeights_eq_zero_of_not_dvd", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/Selberg.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg.jsonl", "positionMetadata": {"lineInFile": 77, 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"Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.NumberTheory.Divisors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.FieldTheory.Tower", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [selbergWeights, if_neg hd]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 38}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\n", "theoremStatement": "lemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b ", "theoremName": "Function.support_deriv_subset_Icc", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": 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"Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 168}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc := by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]\n\nlemma intervalIntegral.norm_integral_le_of_norm_le_const' {a b C : \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {f : \u211d \u2192 E} (hab : a \u2264 b) (h : \u2200 x \u2208 (Icc a b), \u2016f x\u2016 \u2264 C) :\n    \u2016\u222b x in a..b, f x\u2016 \u2264 C * |b - a| := by\n  apply intervalIntegral.norm_integral_le_of_norm_le_const\n  exact fun x hx \u21a6 h x <| mem_Icc_of_Ioc <| uIoc_of_le hab \u25b8 hx\n\n", "theoremStatement": "lemma Filter.TendstoAtZero_of_support_in_Icc {a b : \u211d} (f: \u211d \u2192 \ud835\udd42) (ha : 0 < a)\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    Tendsto f (\ud835\udcdd[>]0) (\ud835\udcdd 0) ", "theoremName": "Filter.TendstoAtZero_of_support_in_Icc", "fileCreated": {"commit": "f36a520e5590b5574377f68ac5fa461e226c424d", "date": "2024-01-22"}, "theoremCreated": {"commit": "806a03bc5bddb48c2e99ef28fd0ec1ad464e93b5", "date": "2024-03-30"}, "file": 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"Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply Tendsto.comp (tendsto_nhds_of_eventually_eq ?_) tendsto_id\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' ha] with c hc; replace hc := (mem_Ioo.mp hc).2\n  have h : c \u2209 Icc a b := fun h \u21a6 by linarith [mem_Icc.mp h]\n  convert mt (Function.support_subset_iff.mp fSupp c) h; simp", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 284}}
+{"srcContext": "import Mathlib.Analysis.MellinInversion\nimport PrimeNumberTheoremAnd.PerronFormula\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\n-- TODO: move near `MeasureTheory.set_integral_prod`\ntheorem MeasureTheory.set_integral_integral_swap {\u03b1 : Type*} {\u03b2 : Type*} {E : Type*}\n    [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] {\u03bc : MeasureTheory.Measure \u03b1}\n    {\u03bd : MeasureTheory.Measure \u03b2} [NormedAddCommGroup E] [MeasureTheory.SigmaFinite \u03bd]\n    [NormedSpace \u211d E] [MeasureTheory.SigmaFinite \u03bc] (f : \u03b1 \u2192 \u03b2 \u2192 E) {s : Set \u03b1} {t : Set \u03b2}\n    (hf : IntegrableOn (f.uncurry) (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)) :\n    (\u222b (x : \u03b1) in s, \u222b (y : \u03b2) in t, f x y \u2202\u03bd \u2202\u03bc)\n      = \u222b (y : \u03b2) in t, \u222b (x : \u03b1) in s, f x y \u2202\u03bc \u2202\u03bd := by\n  apply integral_integral_swap\n  convert hf.integrable\n  exact Measure.prod_restrict s t\n\n-- How to deal with this coersion?... Ans: (f \u00b7)\n--- noncomputable def funCoe (f : \u211d \u2192 \u211d) : \u211d \u2192 \u2102 := fun x \u21a6 f x\n\nopen Complex Topology Filter Real MeasureTheory Set\n\nvariable {\ud835\udd42 : Type*} [RCLike \ud835\udd42]\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_mul_right_Ioi (fun y \u21a6 f y / y) 0 ha\n  simp only [RCLike.ofReal_mul, zero_mul, eq_inv_smul_iff\u2080 (ne_of_gt ha)] at this\n  rw [\u2190 integral_smul] at this\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro _ _\n  simp only [RCLike.real_smul_eq_coe_mul]\n  rw [mul_comm (a : \ud835\udd42), div_mul, mul_div_assoc, div_self ?_, mul_one]\n  exact (RCLike.ofReal_ne_zero).mpr <| ne_of_gt ha\n\nlemma MeasureTheory.integral_comp_mul_right_I0i_haar_real\n    (f : \u211d \u2192 \u211d) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (y * a) / y = \u222b (y : \u211d) in Ioi 0, f y / y :=\n  MeasureTheory.integral_comp_mul_right_I0i_haar f ha\n\nlemma MeasureTheory.integral_comp_mul_left_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a) :\n    \u222b (y : \u211d) in Ioi 0, f (a * y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  convert integral_comp_mul_right_I0i_haar f ha using 5; ring\n\n-- TODO: generalize to `RCLike`\nlemma MeasureTheory.integral_comp_rpow_I0i_haar_real (f : \u211d \u2192 \u211d) {p : \u211d} (hp : p \u2260 0) :\n    \u222b (y : \u211d) in Ioi 0, |p| * f (y ^ p) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  rw [\u2190 integral_comp_rpow_Ioi (fun y \u21a6 f y / y) hp, set_integral_congr (by simp)]\n  intro y hy\n  have ypos : 0 < y := mem_Ioi.mp hy\n  field_simp [rpow_sub_one]\n  ring\n\nlemma MeasureTheory.integral_comp_inv_I0i_haar (f : \u211d \u2192 \ud835\udd42) :\n    \u222b (y : \u211d) in Ioi 0, f (1 / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  have := integral_comp_rpow_Ioi (fun y \u21a6 f y / y) (p := -1) (by simp)\n  rw [\u2190 this, set_integral_congr (by simp)]\n  intro y hy\n  have : (y : \ud835\udd42) \u2260 0 := (RCLike.ofReal_ne_zero).mpr <| LT.lt.ne' hy\n  field_simp [RCLike.real_smul_eq_coe_mul]\n  ring_nf\n  rw [rpow_neg_one, mul_assoc, rpow_neg <| le_of_lt <| mem_Ioi.mp hy]\n  field_simp [pow_two]\n\nlemma MeasureTheory.integral_comp_div_I0i_haar\n    (f : \u211d \u2192 \ud835\udd42) {a : \u211d} (ha : 0 < a):\n    \u222b (y : \u211d) in Ioi 0, f (a / y) / y = \u222b (y : \u211d) in Ioi 0, f y / y := by\n  calc\n    _ = \u222b (y : \u211d) in Ioi 0, f (a * y) / y := ?_\n    _ = _ := integral_comp_mul_left_I0i_haar f ha\n  convert (integral_comp_inv_I0i_haar fun y \u21a6 f (a * (1 / y))).symm using 4\n  \u00b7 rw [mul_one_div]\n  \u00b7 rw [one_div_one_div]\n\ntheorem Complex.ofReal_rpow {x : \u211d} (h : x > 0) (y : \u211d) :\n    (((x : \u211d) ^ (y : \u211d)) : \u211d) = (x : \u2102) ^ (y : \u2102) := by\n  rw [rpow_def_of_pos h, ofReal_exp, ofReal_mul, Complex.ofReal_log h.le,\n    Complex.cpow_def_of_ne_zero]\n  simp only [ne_eq, ofReal_eq_zero, ne_of_gt h, not_false_eq_true]\n\n@[simp]\nlemma Function.support_abs {\u03b1 : Type*} (f : \u03b1 \u2192 \ud835\udd42):\n    (fun x \u21a6 \u2016f x\u2016).support = f.support := by\n  simp only [support, ne_eq, mem_setOf_eq]; simp_rw [norm_ne_zero_iff]\n\n@[simp]\nlemma Function.support_ofReal {f : \u211d \u2192 \u211d} :\n    (fun x \u21a6 ((f x) : \u2102)).support = f.support := by\n  apply Function.support_comp_eq (g := ofReal'); simp [ofReal_zero]\n\nlemma Function.support_id : Function.support (fun x : \u211d \u21a6 x) = Iio 0 \u222a Ioi 0 := by\n  ext x; simp only [mem_support, ne_eq, Iio_union_Ioi, mem_compl_iff, mem_singleton_iff]\n\nlemma Function.support_mul_subset_of_subset {s : Set \u211d} {f g : \u211d \u2192 \ud835\udd42} (fSupp : f.support \u2286 s) :\n    (f * g).support \u2286 s := by\n  simp_rw [support_mul', inter_subset, subset_union_of_subset_right fSupp]\n\nlemma Function.support_of_along_fiber_subset_subset {\u03b1 \u03b2 M : Type*} [Zero M]\n    {f : \u03b1 \u00d7 \u03b2 \u2192 M} {s : Set \u03b1} {t : Set \u03b2}\n    (hx : \u2200 (y : \u03b2), (fun x \u21a6 f (x, y)).support \u2286 s)\n    (hy : \u2200 (x : \u03b1), (fun y \u21a6 f (x, y)).support \u2286 t) :\n    f.support \u2286 s \u00d7\u02e2 t := by\n  intro \u27e8x, y\u27e9 hxy\n  constructor\n  \u00b7 exact hx y (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n  \u00b7 exact hy x (by simp only [Function.mem_support, ne_eq] at hxy \u22a2; exact hxy)\n\nlemma Function.support_deriv_subset_Icc {a b : \u211d} {f : \u211d \u2192 \ud835\udd42}\n    (fSupp : f.support \u2286 Set.Icc a b) :\n    (deriv f).support \u2286 Set.Icc a b := by\n    have := support_deriv_subset (f := fun x \u21a6 f x)\n    dsimp [tsupport] at this\n    have := subset_trans this <| closure_mono fSupp\n    rwa [closure_Icc] at this\n\nlemma IntervalIntegral.integral_eq_integral_of_support_subset_Icc {a b : \u211d} {\u03bc : Measure \u211d} [NoAtoms \u03bc]\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E] [CompleteSpace E]\n    {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) :\n    \u222b x in a..b, f x \u2202\u03bc = \u222b x, f x \u2202\u03bc := by\n  rcases le_total a b with hab | hab\n  \u00b7 rw [intervalIntegral.integral_of_le hab, \u2190 integral_Icc_eq_integral_Ioc,\n    \u2190 integral_indicator measurableSet_Icc, indicator_eq_self.2 h]\n  \u00b7 by_cases hab2 : b = a\n    \u00b7 rw [hab2] at h \u22a2\n      simp [intervalIntegral.integral_same]\n      simp only [Icc_self] at h\n      have : \u222b (x : \u211d), f x \u2202\u03bc = \u222b (x : \u211d) in {a}, f x \u2202\u03bc := by\n        rw [ \u2190 integral_indicator (by simp), indicator_eq_self.2 h]\n      rw [this, integral_singleton]; simp\n    \u00b7 have : \u00aca \u2264 b := by exact fun x \u21a6 hab2 <| le_antisymm hab x\n      rw [Icc_eq_empty_iff.mpr <| by exact fun x \u21a6 hab2 <| le_antisymm hab x, subset_empty_iff,\n          Function.support_eq_empty_iff] at h; simp [h]\n\nlemma SetIntegral.integral_eq_integral_inter_of_support_subset {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s t : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 t) (ht : MeasurableSet t):\n    \u222b x in s, f x \u2202\u03bc = \u222b x in s \u2229 t, f x \u2202\u03bc := by\n  rw [\u2190 set_integral_indicator ht, indicator_eq_self.2 h]\n\n", "theoremStatement": "lemma SetIntegral.integral_eq_integral_inter_of_support_subset_Icc {a b} {\u03bc : Measure \u211d}\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace \u211d E]\n    {s : Set \u211d} {f : \u211d \u2192 E} (h : f.support \u2286 Icc a b) (hs : Icc a b \u2286 s) :\n    \u222b x in s, f x \u2202\u03bc = \u222b x in Icc a b, f x \u2202\u03bc ", "theoremName": 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"Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.Fourier.Inversion", "Mathlib.Analysis.MellinInversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener", "PrimeNumberTheoremAnd.PerronFormula"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [SetIntegral.integral_eq_integral_inter_of_support_subset h measurableSet_Icc,\n      inter_eq_self_of_subset_right hs]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 129}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Basic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.Wiener\n\nopen Asymptotics Complex ComplexConjugate Topology Filter Real MeasureTheory Set\n\nopen scoped Interval\n\n-- TODO: why do we need to bump this?\ninstance : MeasurableDiv\u2082 \u211d := by\n  haveI (G : Type) [DivInvMonoid G] [MeasurableSpace G] [MeasurableInv G] [MeasurableMul\u2082 G] :\n    MeasurableDiv\u2082 G := inferInstance\n  exact this \u211d\n\n/-%%\nIn this section, we prove the Perron formula, which plays a key role in our proof of Mellin inversion.\n%%-/\n\n/-%%\nThe following is preparatory material used in the proof of the Perron formula, see Lemma \\ref{formulaLtOne}.\n%%-/\n\n/-%\nTODO: move to general section.\n\\begin{lemma}[zeroTendstoDiff]\\label{zeroTendstoDiff}\\lean{zeroTendstoDiff}\\leanok\nIf the limit of $0$ is $L\u2081 - L\u2082$, then $L\u2081 = L\u2082$.\n\\end{lemma}\n%-/\nlemma zeroTendstoDiff (L\u2081 L\u2082 : \u2102) (f : \u211d \u2192 \u2102) (h : \u2200\u1da0 T in atTop,  f T = 0)\n    (h' : Tendsto f atTop (\ud835\udcdd (L\u2082 - L\u2081))) : L\u2081 = L\u2082 := by\n  rw [\u2190 zero_add L\u2081, \u2190 @eq_sub_iff_add_eq]\n  exact tendsto_nhds_unique (EventuallyEq.tendsto h) h'\n/-%\n\\begin{proof}\\leanok\nObvious.\n\\end{proof}\n%-/\n\n/-%\nTODO: Move this to general section.\n\\begin{lemma}[RectangleIntegral_tendsTo_VerticalIntegral]\\label{RectangleIntegral_tendsTo_VerticalIntegral}\\lean{RectangleIntegral_tendsTo_VerticalIntegral}\\leanok\n\\uses{RectangleIntegral}\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\lim_{T\\to\\infty}\\int_{\\sigma-iT}^{\\sigma'+iT}f(s)ds =\n\\int_{(\\sigma')}f(s)ds - \\int_{(\\sigma)}f(s)ds.$$\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_VerticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (T : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) atTop\n      (\ud835\udcdd (VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3)) := by\n/-%\n\\begin{proof}\\leanok\nAlmost by definition.\n%-/\n  simp only [RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im, ofReal_im,\n    mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add, zero_sub,\n    add_im]\n  apply Tendsto.sub\n  \u00b7 rewrite [\u2190 zero_add (VerticalIntegral _ _), \u2190 zero_sub_zero]\n    apply Tendsto.add <| Tendsto.sub (hbot.comp tendsto_neg_atTop_atBot) htop\n    exact (intervalIntegral_tendsto_integral hright tendsto_neg_atTop_atBot tendsto_id).const_smul I\n  \u00b7 exact (intervalIntegral_tendsto_integral hleft tendsto_neg_atTop_atBot tendsto_id).const_smul I\n--%\\end{proof}\n\nlemma verticalIntegral_eq_verticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hf : HolomorphicOn f ([[\u03c3,  \u03c3']] \u00d7\u2102 univ))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    VerticalIntegral f \u03c3 = VerticalIntegral f \u03c3' := by\n  refine zeroTendstoDiff _ _ _ (univ_mem' fun _ \u21a6 ?_)\n    (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  exact integral_boundary_rect_eq_zero_of_differentiableOn f _ _\n    (hf.mono fun z hrect \u21a6 \u27e8by simpa using hrect.1, trivial\u27e9)\n\nlemma verticalIntegral_sub_verticalIntegral_eq_squareIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102} {p : \u2102}\n    (h\u03c3: \u03c3 < p.re \u2227 p.re < \u03c3') (hf : HolomorphicOn f (Icc \u03c3  \u03c3' \u00d7\u2102 univ \\ {p}))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3 =\n    RectangleIntegral f (-c - c * I + p) (c + c * I + p) := by\n  have : Icc \u03c3 \u03c3' \u00d7\u2102 univ \u2208 \ud835\udcdd p := by\n    rw [\u2190 mem_interior_iff_mem_nhds, Complex.interior_reProdIm, interior_Icc, interior_univ]\n    refine \u27e8\u27e8?_, ?_\u27e9, trivial\u27e9 <;> linarith\n  obtain \u27e8c', hc'0, hc'\u27e9 := ((nhds_hasBasis_square p).1 _).mp this\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' hc'0] with c \u27e8hc0, hcc'\u27e9\n  have hsub : Square p c \u2286 Icc \u03c3 \u03c3' \u00d7\u2102 univ := (square_subset_square hc0 hcc'.le).trans hc'\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  apply Filter.EventuallyEq.tendsto\n  filter_upwards [Filter.Ioi_mem_atTop ((c - p.im) \u2294 (c + p.im))] with y hy\n  have : c - p.im < y \u2227 c + p.im < y := sup_lt_iff.mp hy\n  have : c + \u03c3 \u2264 p.re := by simpa using (hsub \u27e8left_mem_uIcc, left_mem_uIcc\u27e9).1.1\n  have : c + p.re \u2264 \u03c3' := by simpa using (hsub \u27e8right_mem_uIcc, right_mem_uIcc\u27e9).1.2\n  apply RectanglePullToNhdOfPole'\n  \u00b7 simpa using \u27e8by linarith, by linarith, by linarith\u27e9\n  \u00b7 exact square_mem_nhds p (ne_of_gt hc0)\n  \u00b7 apply RectSubRect' <;> simpa using by linarith\n  \u00b7 refine hf.mono (diff_subset_diff ?_ subset_rfl)\n    simpa [Rectangle, uIcc_of_lt (h\u03c3.1.trans h\u03c3.2)] using fun x \u27e8hx, _\u27e9 \u21a6 \u27e8hx, trivial\u27e9\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_UpperU]\\label{RectangleIntegral_tendsTo_UpperU}\\lean{RectangleIntegral_tendsTo_UpperU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma+iT}^{\\sigma'+iU}f(s)ds$$\nas $U\\to\\infty$ is the ``UpperUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_UpperU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 + I * T) (\u03c3' + I * U)) atTop\n      (\ud835\udcdd (UpperUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, UpperUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  + I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  + I * t).im = t  := by simp\n  have hbot : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) := by\n    exact tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in T..U, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Ioi T, f (s + y * I)) := by\n    exact (intervalIntegral_tendsto_integral_Ioi T int.restrict tendsto_id).const_smul I\n  have := ((hbot.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  simpa only [RectangleIntegral, UpperUIntegral, h_re, h_im, sub_zero, \u2190integral_Ici_eq_integral_Ioi]\n--%\\end{proof}\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_LowerU]\\label{RectangleIntegral_tendsTo_LowerU}\\lean{RectangleIntegral_tendsTo_LowerU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to -\\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma-iU}^{\\sigma'-iT}f(s)ds$$\nas $U\\to\\infty$ is the ``LowerUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_LowerU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * U) (\u03c3' - I * T)) atTop\n      (\ud835\udcdd (- LowerUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, LowerUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  - I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  - I * t).im = -t  := by simp\n  have hbot' : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - y * I)) atTop (\ud835\udcdd 0) := by\n    convert (hbot.comp tendsto_neg_atTop_atBot) using 1\n    ext; simp only [Function.comp_apply, ofReal_neg, neg_mul]; rfl\n  have htop : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) :=\n    tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in -U..-T, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Iic (-T), f (s + y * I)) := by\n    have := (intervalIntegral_tendsto_integral_Iic (-T) int.restrict tendsto_id).const_smul I\n    convert (this.comp tendsto_neg_atTop_atBot) using 1\n  have := ((hbot'.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  rw [zero_sub] at this\n  simp_rw [RectangleIntegral, LowerUIntegral, HIntegral, VIntegral, h_re, h_im, ofReal_neg, neg_mul, neg_add_rev, neg_sub]\n  have final : (((-\u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I)) + I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) -\n      I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) = (-(I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) +\n      ((I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) - \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I))) := by\n    ring_nf\n  exact final \u25b8 this\n--%\\end{proof}\n\n/-%%\nTODO : Move to general section\n\\begin{lemma}[limitOfConstant]\\label{limitOfConstant}\\lean{limitOfConstant}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma>0$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstant {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : 0 < \u03c3') (_ : 0 < \u03c3''), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atTop (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\\begin{align*}\n\\lim_{\\sigma'\\to\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atTop \u03c3) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 (\u03c3pos.trans_le h) \u03c3pos\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[limitOfConstantLeft]\\label{limitOfConstantLeft}\\lean{limitOfConstantLeft}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma<-3/2$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to-\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstantLeft {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3lt : \u03c3 \u2264 -3/2)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : \u03c3' \u2264 -3/2) (_ : \u03c3'' \u2264 -3/2), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atBot (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\begin{align*}\n\\lim_{\\sigma'\\to-\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to-\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atBot (-3/2)) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 h \u03c3lt\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_lt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\leanok\nLet $x>0$ and $x<1$. Then\n$$\\lim_{\\sigma\\to\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_lt_one {x : \u211d}  (xpos : 0 < x) (x_lt_one : x < 1) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nStandard.\n%%-/\n  have := Tendsto.mul_const C (tendsto_rpow_atTop_of_base_lt_one x (by linarith) x_lt_one)\n  simpa only [rpow_eq_pow, zero_mul] using this\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_gt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\leanok\nLet $x>1$. Then\n$$\\lim_{\\sigma\\to-\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_gt_one {x : \u211d} (x_gt_one : 1 < x) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atBot (\ud835\udcdd 0) := by\n  have := (zero_lt_one.trans x_gt_one)\n  have h := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one (inv_pos.mpr this) (inv_lt_one x_gt_one) C\n  convert (h.comp tendsto_neg_atBot_atTop) using 1\n  ext; simp only [this.le, inv_rpow, Function.comp_apply, rpow_neg, inv_inv]\n\n/-%%\n\\begin{proof}\\leanok\nStandard.\n\\end{proof}\n%%-/\n\n-- TODO: move near `Complex.cpow_neg`?\nlemma Complex.cpow_inv_ofReal_pos {a : \u211d} (ha : 0 \u2264 a) (r : \u2102) :\n    ((a : \u2102) ^ r)\u207b\u00b9 = (a : \u2102)\u207b\u00b9 ^ r := by\n  sorry\n\nlemma Complex.cpow_eq_exp_log_ofReal (x : \u211d) (hx : 0 < x) (y : \u2102) :\n    (x : \u2102) ^ y = Complex.exp (Real.log x * y) := by\n  simp [\u2190 Complex.cpow_eq_pow, Complex.cpow, hx.ne.symm, \u2190 Complex.ofReal_log hx.le]\n\n-- TODO: move near `Complex.mul_cpow_ofReal_nonneg`\nlemma Complex.cpow_neg_eq_inv_pow_ofReal_pos {a : \u211d} (ha : 0 < a) (r : \u2102) :\n    (a : \u2102) ^ (-r) = (a\u207b\u00b9 : \u2102) ^ r := by\n  rw [cpow_neg, \u2190 Complex.inv_cpow]\n  exact slitPlane_arg_ne_pi (Or.inl ha)\n\nnamespace Perron\n\nvariable {x \u03c3 \u03c3' \u03c3'' T : \u211d}\n\nnoncomputable abbrev f (x : \u211d) := fun (s : \u2102) \u21a6 x ^ s / (s * (s + 1))\n\n\nlemma f_mul_eq_f {x t : \u211d} (tpos : 0 < t) (xpos : 0 < x) (s : \u2102) : f t s * (x : \u2102) ^ (-s) = f (t / x) s := by\n  by_cases s_eq_zero : s = 0\n  \u00b7 simp [f, s_eq_zero]\n  by_cases s_eq_neg_one : s = -1\n  \u00b7 simp [f, s_eq_neg_one]\n  field_simp [f, mul_ne_zero s_eq_zero (fun hs \u21a6 add_eq_zero_iff_eq_neg.mp hs |> s_eq_neg_one)]\n  convert (Complex.mul_cpow_ofReal_nonneg tpos.le (inv_pos.mpr xpos).le s).symm using 2\n  \u00b7 convert Complex.cpow_neg_eq_inv_pow_ofReal_pos xpos s\n    exact ofReal_inv x\n  \u00b7 simp only [ofReal_inv]; rfl\n\n/-%%\n\\begin{lemma}[isHolomorphicOn]\\label{isHolomorphicOn}\\lean{Perron.isHolomorphicOn}\\leanok\nLet $x>0$. Then the function $f(s) = x^s/(s(s+1))$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n\\end{lemma}\n%%-/\nlemma isHolomorphicOn (xpos : 0 < x) : HolomorphicOn (f x) {0, -1}\u1d9c := by\n/-%%\n\\begin{proof}\\leanok\nComposition of differentiabilities.\n%%-/\n  unfold f\n  simp_rw [Complex.cpow_def_of_ne_zero <| ofReal_ne_zero.mpr <| ne_of_gt xpos]\n  apply DifferentiableOn.div <| DifferentiableOn.cexp <| DifferentiableOn.const_mul differentiableOn_id _\n  \u00b7 exact DifferentiableOn.mul differentiableOn_id <| DifferentiableOn.add_const differentiableOn_id 1\n  \u00b7 intro x hx\n    obtain \u27e8h0, h1\u27e9 := not_or.mp hx\n    exact mul_ne_zero h0 <| add_ne_add_left 1 |>.mpr h1 |>.trans_eq (add_left_neg 1)\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[integralPosAux]\\label{integralPosAux}\\lean{Perron.integralPosAux}\\leanok\nThe integral\n$$\\int_\\R\\frac{1}{|(1+t^2)(2+t^2)|^{1/2}}dt$$\nis positive (and hence convergent - since a divergent integral is zero in Lean, by definition).\n\\end{lemma}\n%%-/\n\nlemma integral_one_div_const_add_sq_pos (c : \u211d) (hc : 0 < c) : 0 < \u222b (t : \u211d), 1 / (c + t ^ 2) := by\n  have hfun_eq (t : \u211d) : 1 / (c + t ^ 2) = c\u207b\u00b9 * (1 + (c.sqrt\u207b\u00b9 * t) ^ 2)\u207b\u00b9 := by\n    field_simp [hc.ne.symm]\n  simp_rw [hfun_eq, MeasureTheory.integral_mul_left,\n    Measure.integral_comp_mul_left (fun t \u21a6 (1 + t ^ 2)\u207b\u00b9) (a:=c.sqrt\u207b\u00b9)]\n  simp [abs_eq_self.mpr <| Real.sqrt_nonneg c,\n    mul_pos (inv_pos.mpr hc) <| mul_pos (sqrt_pos.mpr hc) Real.pi_pos]\n\nlemma Integrable.one_div_const_add_sq (c : \u211d) (hc : 0 < c) : Integrable fun (t : \u211d) \u21a6 1 / (c + t ^ 2) :=\n  .of_integral_ne_zero (integral_one_div_const_add_sq_pos c hc).ne'\n\nlemma integralPosAux'_of_le (c\u2081 c\u2082 : \u211d) (c\u2081_pos : 0 < c\u2081) (hle : c\u2081 \u2264 c\u2082) :\n    0 < \u222b (t : \u211d), 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) := by\n  have c\u2082_pos : 0 < c\u2082 := by linarith\n  have hlower (t : \u211d) : 1 / (c\u2082 + t ^ 2) \u2264 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) := by\n    gcongr\n    calc\n      _ \u2264 (c\u2082 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt := by gcongr; apply Real.sqrt_le_sqrt; gcongr\n      _ \u2264 c\u2082 + t ^ 2 := by rw [\u2190 Real.sqrt_mul, sqrt_mul_self] <;> positivity\n  have hupper (t : \u211d) : 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) \u2264 1 / (c\u2081 + t ^ 2)  := by\n      gcongr\n      calc\n        _ \u2265 (c\u2081 + t ^ 2).sqrt * (c\u2081 + t ^ 2).sqrt := by gcongr; apply Real.sqrt_le_sqrt; gcongr\n        _ \u2265 c\u2081 + t ^ 2 := by rw [\u2190 Real.sqrt_mul, sqrt_mul_self] <;> positivity\n  calc 0 < \u222b t, 1 / (c\u2082 + t^2) := integral_one_div_const_add_sq_pos c\u2082 c\u2082_pos\n       _ \u2264 \u222b t, 1 / (Real.sqrt (c\u2081 + t^2) * Real.sqrt (c\u2082 + t^2)) := ?_\n  refine integral_mono (Integrable.one_div_const_add_sq c\u2082 c\u2082_pos) ?_ hlower\n  apply MeasureTheory.Integrable.mono (g := fun t:\u211d \u21a6 1/(c\u2081 + t^2)) <| Integrable.one_div_const_add_sq c\u2081 c\u2081_pos\n  \u00b7 refine (measurable_const.div <| Measurable.mul ?_ ?_).aestronglyMeasurable <;>\n      exact (measurable_const.add <| measurable_id'.pow_const 2).sqrt\n  \u00b7 refine ae_of_all _ (fun x \u21a6 ?_)\n    repeat rewrite [norm_of_nonneg (by positivity)]\n    exact hupper x\n\n\nlemma integralPosAux' (c\u2081 c\u2082 : \u211d) (c\u2081_pos : 0 < c\u2081) (c\u2082_pos : 0 < c\u2082) :\n    0 < \u222b (t : \u211d), 1 / ((c\u2081 + t^2).sqrt * (c\u2082 + t^2).sqrt) := by\n  by_cases hc : c\u2081 \u2264 c\u2082\n  \u00b7 exact integralPosAux'_of_le c\u2081 c\u2082 c\u2081_pos hc\n  \u00b7 convert integralPosAux'_of_le c\u2082 c\u2081 c\u2082_pos (by linarith) using 4; rw [mul_comm]\n\nlemma integralPosAux : 0 < \u222b (t : \u211d), 1 / ((1 + t^2).sqrt * (2 + t^2).sqrt) := by\n/-%%\n\\begin{proof}\\leanok\nThis integral is between $\\frac{1}{2}$ and $1$ of the integral of $\\frac{1}{1+t^2}$, which is $\\pi$.\n%%-/\n  apply integralPosAux' <;> norm_num\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[vertIntBound]\\label{vertIntBound}\\lean{Perron.vertIntBound}\\leanok\nLet $x>0$ and $\\sigma>1$. Then\n$$\\left|\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds\\right| \\leq x^\\sigma \\int_\\R\\frac{1}{|(1+t ^ 2)(2+t ^ 2)|^{1/2}}dt.$$\n\\end{lemma}\n%%-/\nlemma vertIntBound (xpos : 0 < x) (\u03c3_gt_one : 1 < \u03c3) :\n    \u2016VerticalIntegral (f x) \u03c3\u2016 \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt) := by\n  calc\n    _ = \u2016\u222b (t : \u211d), x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ \u2264 \u222b (t : \u211d), \u2016x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 :=\n        norm_integral_le_integral_norm _\n    _ = \u222b (t : \u211d), x ^ \u03c3 / \u2016((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ = x ^ \u03c3 * \u222b (t : \u211d), 1 / (\u2016\u03c3 + t * I\u2016 * \u2016\u03c3 + t * I + 1\u2016) := ?_\n    _ \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt) :=\n        mul_le_mul_of_nonneg_left ?_ (rpow_nonneg xpos.le _)\n  \u00b7 simp [VerticalIntegral]\n  \u00b7 simp [Complex.abs_cpow_eq_rpow_re_of_pos xpos]\n  \u00b7 simp [integral_mul_left, div_eq_mul_inv]\n  by_cases hint : Integrable fun (a : \u211d) \u21a6 1 / (\u2016\u03c3 + a * I\u2016 * \u2016\u03c3 + a * I + 1\u2016)\n  swap; rw [integral_undef hint]; exact integral_nonneg <| fun t \u21a6 by positivity\n  conv => rhs; rhs; intro a; rhs\n  apply integral_mono hint\n  \u00b7 have := integralPosAux\n    contrapose! this\n    simp_rw [integral_undef this, le_rfl]\n  rw [Pi.le_def]\n  intro t\n  gcongr <;> apply sqrt_le_sqrt\n  \u00b7 simp_rw [normSq_add_mul_I, add_le_add_iff_right, one_le_pow_of_one_le \u03c3_gt_one.le _]\n  \u00b7 rw [add_right_comm, \u2190 ofReal_one, \u2190 ofReal_add, normSq_add_mul_I, add_le_add_iff_right]\n    nlinarith\n  rfl\n/-%%\n\\begin{proof}\\leanok\n\\uses{VerticalIntegral}\nTriangle inequality and pointwise estimate.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[vertIntBoundLeft]\\label{vertIntBoundLeft}\\lean{Perron.vertIntBoundLeft}\\leanok\nLet $x>1$ and $\\sigma<-3/2$. Then\n$$\\left|\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds\\right| \\leq x^\\sigma \\int_\\R\\frac{1}{|(1/4+t ^ 2)(2+t ^ 2)|^{1/2}}dt.$$\n\\end{lemma}\n%%-/\n\nlemma vertIntBoundLeft (xpos : 0 < x) :\n    \u2203 C, \u2200 (\u03c3 : \u211d) (_ : \u03c3 < -3 / 2), \u2016VerticalIntegral' (f x) \u03c3\u2016 \u2264 C * x ^ \u03c3 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{VerticalIntegral}\n%%-/\n  /- This proof is adapted from `vertIntBound` -/\n  use 1 / (2 * \u03c0) *  \u2016(\u222b (t : \u211d), 1 / ((4\u207b\u00b9 + t ^ 2).sqrt * (4\u207b\u00b9 + t ^ 2).sqrt : \u2102))\u2016\n  intro \u03c3 h\u03c3\n  simp only [VerticalIntegral', abs_of_pos Real.pi_pos, smul_eq_mul, norm_mul, f]\n  rw [(by simp [pi_nonneg] : \u20161 / (2 * \u2191\u03c0 * I)\u2016 = 1 / (2 * \u03c0)), mul_assoc]\n  apply (mul_le_mul_left (by simp [pi_pos])).mpr\n  calc\n    _ = \u2016\u222b (t : \u211d), x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ \u2264 \u222b (t : \u211d), \u2016x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := norm_integral_le_integral_norm _\n    _ = \u222b (t : \u211d), x ^ \u03c3 / \u2016((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ = x ^ \u03c3 * \u222b (t : \u211d), 1 / (\u2016\u03c3 + t * I\u2016 * \u2016\u03c3 + t * I + 1\u2016) := ?_\n    _ \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((4\u207b\u00b9 + t ^ 2).sqrt * (4\u207b\u00b9 + t ^ 2).sqrt) := ?_\n    _ \u2264 _ := ?_\n  \u00b7 simp [VerticalIntegral, Real.pi_nonneg]\n  \u00b7 congr with t\n    rw [norm_div, Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos xpos, add_re, ofReal_re,\n      re_ofReal_mul, I_re, mul_zero, add_zero]\n  \u00b7 simp_rw [div_eq_mul_inv, integral_mul_left, one_mul, Complex.norm_eq_abs, map_mul]\n  \u00b7 gcongr\n    by_cases hint : Integrable fun (a : \u211d) \u21a6 1 / (\u2016\u03c3 + \u2191a * I\u2016 * \u2016\u03c3 + \u2191a * I + 1\u2016)\n    swap\n    \u00b7 rw [integral_undef hint]\n      exact integral_nonneg <| fun t \u21a6 by simp only [Pi.le_def, Pi.zero_apply]; positivity\n    apply integral_mono hint\n    \u00b7 have := integralPosAux' (4\u207b\u00b9) (4\u207b\u00b9) (by norm_num) (by norm_num)\n      contrapose! this\n      simp_rw [integral_undef this, le_rfl]\n    rw [Pi.le_def]\n    intro t\n    gcongr <;> apply sqrt_le_sqrt\n    \u00b7 rw [normSq_add_mul_I, add_le_add_iff_right]; ring_nf; nlinarith\n    \u00b7 rw [(by push_cast; ring : \u03c3 + t * I + 1 = ofReal' (\u03c3 + 1) + t * I),\n        normSq_add_mul_I, add_le_add_iff_right]; ring_nf; nlinarith\n  \u00b7 rw [mul_comm]\n    gcongr\n    \u00b7 have : 0 \u2264 \u222b (t : \u211d), 1 / (sqrt (4\u207b\u00b9 + t ^ 2) * sqrt (4\u207b\u00b9 + t ^ 2)) := by positivity\n      rw [\u2190 _root_.abs_of_nonneg this, \u2190 Complex.abs_ofReal]\n      apply le_of_eq; congr; norm_cast; exact integral_ofReal.symm\n/-%%\nTriangle inequality and pointwise estimate.\n\\end{proof}\n%%-/\n\nlemma map_conj (hx : 0 \u2264 x) (s : \u2102) : f x (conj s) = conj (f x s) := by\n  simp only [f, map_div\u2080, map_mul, map_add, map_one]\n  congr\n  rw [cpow_conj, Complex.conj_ofReal]; rw [Complex.arg_ofReal_of_nonneg hx]; exact pi_ne_zero.symm\n\ntheorem isTheta_uniformlyOn_uIcc {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    (fun (\u03c3, (y : \u211d)) \u21a6 f x (\u03c3 + y * I)) =\u0398[\ud835\udcdf [[\u03c3', \u03c3'']] \u00d7\u02e2 (atBot \u2294 atTop)]\n    ((fun y \u21a6 1 / y^2) \u2218 Prod.snd) := by\n  set l := \ud835\udcdf [[\u03c3', \u03c3'']] \u00d7\u02e2 (atBot \u2294 atTop : Filter \u211d) with hl\n  refine IsTheta.div (isTheta_norm_left.mp ?_) ?_\n  \u00b7 suffices (fun (\u03c3, _y) \u21a6 |x| ^ \u03c3) =\u0398[l] fun _ \u21a6 (1 : \u211d) by\n      simpa [Complex.abs_cpow_of_ne_zero <| ofReal_ne_zero.mpr (ne_of_gt xpos),\n        arg_ofReal_of_nonneg xpos.le] using this\n    exact (continuousOn_const.rpow continuousOn_id fun _ _ \u21a6 Or.inl <| ne_of_gt (abs_pos_of_pos xpos))\n      |>.const_isThetaUniformlyOn_isCompact isCompact_uIcc (by norm_num)\n      (fun i _ \u21a6 ne_of_gt <| rpow_pos_of_pos (abs_pos_of_pos xpos) _) _\n  \u00b7 have h_c {c : \u2102} : (fun (_ : \u211d \u00d7 \u211d) \u21a6 c) =o[l] Prod.snd := by\n      rewrite [hl, Filter.prod_sup, isLittleO_sup]\n      exact \u27e8isLittleO_const_snd_atBot c _, isLittleO_const_snd_atTop c _\u27e9\n    have h_yI : (fun ((_\u03c3, y) : \u211d \u00d7 \u211d) \u21a6 y * I) =\u0398[l] Prod.snd :=\n      isTheta_of_norm_eventuallyEq (by simp)\n    have h_\u03c3_yI : (fun (\u03c3y : \u211d \u00d7 \u211d) \u21a6 \u03c3y.1 + \u03c3y.2 * I) =\u0398[l] Prod.snd := by\n      refine IsLittleO.add_isTheta ?_ h_yI\n      exact continuous_ofReal.continuousOn.const_isBigOUniformlyOn_isCompact isCompact_uIcc\n        (by norm_num : \u2016(1 : \u2102)\u2016 \u2260 0) _ |>.trans_isLittleO h_c\n    simp_rw [sq]; exact h_\u03c3_yI.mul (h_\u03c3_yI.add_isLittleO h_c)\n\ntheorem isTheta_uniformlyOn_uIoc {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    (fun (\u03c3, (y : \u211d)) \u21a6 f x (\u03c3 + y * I)) =\u0398[\ud835\udcdf (uIoc \u03c3' \u03c3'') \u00d7\u02e2 (atBot \u2294 atTop)]\n    fun (\u03c3, y) \u21a6 1 / y^2 := by\n  refine (\ud835\udcdf (uIoc \u03c3' \u03c3'')).eq_or_neBot.casesOn (fun hbot \u21a6 by simp [hbot]) (fun _ \u21a6 ?_)\n  haveI : NeBot (atBot (\u03b1 := \u211d) \u2294 atTop) := sup_neBot.mpr (Or.inl atBot_neBot)\n  exact (isTheta_uniformlyOn_uIcc xpos \u03c3' \u03c3'').mono (by simpa using Ioc_subset_Icc_self)\n\nlemma isTheta (xpos : 0 < x) :\n    ((fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) =\u0398[atBot] fun (y : \u211d) \u21a6 1 / y^2) \u2227\n    (fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) =\u0398[atTop] fun (y : \u211d) \u21a6 1 / y^2 :=\n  isTheta_sup.mp <| isTheta_of_isThetaUniformly (isTheta_uniformlyOn_uIcc xpos \u03c3 \u03c3) left_mem_uIcc\n\n/-%%\n\\begin{lemma}[isIntegrable]\\label{isIntegrable}\\lean{Perron.isIntegrable}\\leanok\nLet $x>0$ and $\\sigma\\in\\R$. Then\n$$\\int_{\\R}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\nis integrable.\n\\end{lemma}\n%%-/\nlemma isIntegrable (xpos : 0 < x) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_neg_one : \u03c3 \u2260 -1) :\n    Integrable fun (t : \u211d) \u21a6 f x (\u03c3 + t * I) := by\n/-%%\n\\begin{proof}\\uses{isHolomorphicOn}\\leanok\nBy \\ref{isHolomorphicOn}, $f$ is continuous, so it is integrable on any interval.\n%%-/\n  have : Continuous (fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) := by\n    refine (isHolomorphicOn xpos).continuousOn.comp_continuous (by continuity) fun x \u21a6 not_or.mpr ?_\n    simp [Complex.ext_iff, \u03c3_ne_zero, \u03c3_ne_neg_one]\n--%% Also, $|f(x)| = \\Theta(x^{-2})$ as $x\\to\\infty$,\n  refine this.locallyIntegrable.integrable_of_isBigO_atTop_of_norm_eq_norm_neg\n    (univ_mem' fun y \u21a6 ?_) (isTheta xpos).2.isBigO \u27e8Ioi 1, Ioi_mem_atTop 1, ?_\u27e9\n--%% and $|f(-x)| = \\Theta(x^{-2})$ as $x\\to\\infty$.\n  \u00b7 show \u2016f x (\u2191\u03c3 + \u2191y * I)\u2016 = \u2016f x (\u2191\u03c3 + \u2191(-y) * I)\u2016\n    have : (\u2191\u03c3 + \u2191(-y) * I) = conj (\u2191\u03c3 + \u2191y * I) := Complex.ext (by simp) (by simp)\n    simp_rw [this, map_conj xpos.le, Complex.norm_eq_abs, abs_conj]\n--%% Since $g(x) = x^{-2}$ is integrable on $[a,\\infty)$ for any $a>0$, we conclude.\n  \u00b7 refine integrableOn_Ioi_rpow_of_lt (show (-2 : \u211d) < -1 by norm_num)\n      (show (0 : \u211d) < 1 by norm_num) |>.congr_fun (fun y hy \u21a6 ?_) measurableSet_Ioi\n    rw [rpow_neg (show (0 : \u211d) < 1 by norm_num |>.trans hy |>.le), inv_eq_one_div, rpow_two]\n--%%\\end{proof}\n\ntheorem horizontal_integral_isBigO\n    {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) (\u03bc : Measure \u211d) [IsLocallyFiniteMeasure \u03bc] :\n    (fun (y : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + y * I) \u2202\u03bc) =O[atBot \u2294 atTop]\n    fun y \u21a6 1 / y^2 := by\n  let g := fun ((\u03c3, y) : \u211d \u00d7 \u211d) \u21a6 f x (\u03c3 + y * I)\n  calc\n    _ =\u0398[atBot \u2294 atTop] fun (y : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in uIoc \u03c3' \u03c3'', g (\u03c3, y) \u2202\u03bc :=\n        isTheta_of_norm_eventuallyEq <| univ_mem'\n          fun _ \u21a6 intervalIntegral.norm_intervalIntegral_eq _ _ _ _\n    _ =O[atBot \u2294 atTop] _ :=\n      (isTheta_uniformlyOn_uIoc xpos \u03c3' \u03c3'').isBigO.set_integral_isBigO\n        measurableSet_uIoc measure_Ioc_lt_top\n\n/-%%\n\\begin{lemma}[tendsto_zero_Lower]\\label{tendsto_zero_Lower}\\lean{Perron.tendsto_zero_Lower}\\leanok\nLet $x>0$ and $\\sigma',\\sigma''\\in\\R$. Then\n$$\\int_{\\sigma'}^{\\sigma''}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\ngoes to $0$ as $t\\to-\\infty$.\n\\end{lemma}\n%%-/\nlemma tendsto_zero_Lower (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + t * I)) atBot (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nThe numerator is bounded and the denominator tends to infinity.\n\\end{proof}\n%%-/\n  have hcast : (fun (y : \u211d) \u21a6 1 / y ^ 2) =\u1da0[atBot] fun y \u21a6 (-y) ^ (-2 : \u211d) := by\n    filter_upwards [Iic_mem_atBot 0] with y hy using\n      by rw [rpow_neg (neg_nonneg.mpr hy), inv_eq_one_div, rpow_two, neg_sq]\n  exact isBigO_sup.mp (horizontal_integral_isBigO xpos \u03c3' \u03c3'' volume)\n    |>.1.trans_eventuallyEq hcast |>.trans_tendsto\n    <| tendsto_rpow_neg_atTop (by norm_num) |>.comp tendsto_neg_atBot_atTop\n\n/-%%\n\\begin{lemma}[tendsto_zero_Upper]\\label{tendsto_zero_Upper}\\lean{Perron.tendsto_zero_Upper}\\leanok\nLet $x>0$ and $\\sigma',\\sigma''\\in\\R$. Then\n$$\\int_{\\sigma'}^{\\sigma''}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\ngoes to $0$ as $t\\to\\infty$.\n\\end{lemma}\n%%-/\nlemma tendsto_zero_Upper (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + t * I)) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nThe numerator is bounded and the denominator tends to infinity.\n\\end{proof}\n%%-/\n  have hcast : (fun (y : \u211d) \u21a6 1 / y ^ 2) =\u1da0[atTop] fun y \u21a6 y ^ (-2 : \u211d) := by\n    filter_upwards [Ici_mem_atTop 0] with y hy using by rw [rpow_neg hy, inv_eq_one_div, rpow_two]\n  refine isBigO_sup.mp (horizontal_integral_isBigO xpos \u03c3' \u03c3'' volume)\n    |>.2.trans_eventuallyEq hcast |>.trans_tendsto <| tendsto_rpow_neg_atTop (by norm_num)\n\nlemma contourPull {\u03c3' \u03c3'' : \u211d} (xpos : 0 < x) (h\u03c30 : 0 \u2209 [[\u03c3', \u03c3'']]) (h\u03c31 : -1 \u2209 [[\u03c3', \u03c3'']]) :\n    VerticalIntegral (f x) \u03c3' = VerticalIntegral (f x) \u03c3'' := by\n  refine verticalIntegral_eq_verticalIntegral ((isHolomorphicOn xpos).mono ?_)\n    (tendsto_zero_Lower xpos \u03c3' \u03c3'') (tendsto_zero_Upper xpos \u03c3' \u03c3'')\n    (isIntegrable xpos (fun h \u21a6 h\u03c30 (h \u25b8 left_mem_uIcc)) (fun h \u21a6 h\u03c31 (h \u25b8 left_mem_uIcc)))\n    (isIntegrable xpos (fun h \u21a6 h\u03c30 (h \u25b8 right_mem_uIcc)) (fun h \u21a6 h\u03c31 (h \u25b8 right_mem_uIcc)))\n  rintro \u27e8x, y\u27e9 \u27e8hx, hy\u27e9 \u27e8hc | hc\u27e9 <;> simp_all [Complex.ext_iff]\n\n/-%%\nWe are ready for the first case of the Perron formula, namely when $x<1$:\n\\begin{lemma}[formulaLtOne]\\label{formulaLtOne}\\lean{Perron.formulaLtOne}\\leanok\nFor $x>0$, $\\sigma>0$, and $x<1$, we have\n$$\n\\frac1{2\\pi i}\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds =0.\n$$\n\\end{lemma}\n%%-/\nlemma formulaLtOne (xpos : 0 < x) (x_lt_one : x < 1) (\u03c3_pos : 0 < \u03c3)\n    : VerticalIntegral (f x) \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{isHolomorphicOn, HolomorphicOn.vanishesOnRectangle, integralPosAux,\nvertIntBound, limitOfConstant,\ntendsto_rpow_atTop_nhds_zero_of_norm_lt_one,\ntendsto_zero_Lower, tendsto_zero_Upper, isIntegrable}\n  Let $f(s) = x^s/(s(s+1))$. Then $f$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n  The rectangle integral of $f$ with corners $\\sigma-iT$ and $\\sigma+iT$ is zero.\n  The limit of this rectangle integral as $T\\to\\infty$ is $\\int_{(\\sigma')}-\\int_{(\\sigma)}$.\n  Therefore, $\\int_{(\\sigma')}=\\int_{(\\sigma)}$.\n%%-/\n  have h_contourPull (\u03c3' \u03c3'' : \u211d) (\u03c3'pos : 0 < \u03c3') (\u03c3''pos : 0 < \u03c3'') :\n      VerticalIntegral (f x) \u03c3' = VerticalIntegral (f x) \u03c3'' :=\n    contourPull xpos (not_mem_uIcc_of_lt \u03c3'pos \u03c3''pos)\n      (not_mem_uIcc_of_lt (by linarith) (by linarith))\n--%% But we also have the bound $\\int_{(\\sigma')} \\leq x^{\\sigma'} * C$, where\n--%% $C=\\int_\\R\\frac{1}{|(1+t)(1+t+1)|}dt$.\n  have VertIntBound : \u2203 C > 0, \u2200 \u03c3' > 1, \u2016VerticalIntegral (f x) \u03c3'\u2016 \u2264 x^\u03c3' * C := by\n    let C := \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt)\n    exact \u27e8C, integralPosAux, fun _ \u21a6 vertIntBound xpos\u27e9\n--%% Therefore $\\int_{(\\sigma')}\\to 0$ as $\\sigma'\\to\\infty$.\n  have AbsVertIntTendsto : Tendsto (Complex.abs \u2218 (VerticalIntegral (f x))) atTop (\ud835\udcdd 0) := by\n    obtain \u27e8C, _, hC\u27e9 := VertIntBound\n    have := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one xpos x_lt_one C\n    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds this\n    \u00b7 filter_upwards; exact fun _ \u21a6 Complex.abs.nonneg' _\n    \u00b7 filter_upwards [eventually_gt_atTop 1]; exact hC\n  have VertIntTendsto : Tendsto (VerticalIntegral (f x)) atTop (\ud835\udcdd 0) :=\n    tendsto_zero_iff_norm_tendsto_zero.mpr AbsVertIntTendsto\n  --%% So pulling contours gives $\\int_{(\\sigma)}=0$.\n  exact limitOfConstant \u03c3_pos h_contourPull VertIntTendsto\n--%%\\end{proof}\n\n/-%%\nThe second case is when $x>1$.\nHere are some auxiliary lemmata for the second case.\nTODO: Move to more general section\n%%-/\n\ntheorem HolomorphicOn.upperUIntegral_eq_zero {f : \u2102 \u2192 \u2102} {\u03c3 \u03c3' T : \u211d} (h\u03c3 : \u03c3 \u2264 \u03c3')\n    (hf : HolomorphicOn f {z : \u2102 | \u03c3 \u2264 z.re \u2227 z.re \u2264 \u03c3' \u2227 T \u2264 z.im})\n    (htop : Tendsto (fun y : \u211d \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x + \u2191y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3 + \u2191y * I))\n    (hright : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3' + \u2191y * I)) :\n    UpperUIntegral f \u03c3 \u03c3' T = 0 := by\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_UpperU htop hleft hright)\n  apply EventuallyEq.tendsto\n  filter_upwards [eventually_ge_atTop T]\n  refine fun _ hTU \u21a6 hf.vanishesOnRectangle fun _ \u21a6 ?_\n  rw [mem_Rect (by simp [h\u03c3]) (by simp [hTU])]\n  simpa using by tauto\n\ntheorem HolomorphicOn.lowerUIntegral_eq_zero {f : \u2102 \u2192 \u2102} {\u03c3 \u03c3' T : \u211d} (h\u03c3 : \u03c3 \u2264 \u03c3')\n    (hf : HolomorphicOn f {z : \u2102 | \u03c3 \u2264 z.re \u2227 z.re \u2264 \u03c3' \u2227 z.im \u2264 -T})\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3 + \u2191y * I))\n    (hright : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3' + \u2191y * I)) :\n    LowerUIntegral f \u03c3 \u03c3' T = 0 := by\n  suffices h : - LowerUIntegral f \u03c3 \u03c3' T = 0 by exact neg_eq_zero.mp h\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_LowerU hbot hleft hright)\n  apply EventuallyEq.tendsto\n  filter_upwards [eventually_ge_atTop T]\n  refine fun _ hTU \u21a6 hf.vanishesOnRectangle fun _ \u21a6 ?_\n  rw [mem_Rect (by simp [h\u03c3]) (by simp [hTU])]\n  simpa using by tauto\n\n", "theoremStatement": "lemma sPlusOneNeZero {s : \u2102} (s_ne_neg_one : s \u2260 -1) : s + 1 \u2260 0 ", "theoremName": "Perron.sPlusOneNeZero", 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"Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Integral.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  fun h \u21a6 s_ne_neg_one (add_eq_zero_iff_eq_neg.mp h)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 55}}
+{"srcContext": "import EulerProducts.PNT\nimport Mathlib.Analysis.Fourier.FourierTransform\nimport Mathlib.Analysis.Fourier.FourierTransformDeriv\nimport Mathlib.NumberTheory.ArithmeticFunction\nimport Mathlib.Topology.Support\nimport Mathlib.Analysis.Calculus.ContDiff.Defs\nimport Mathlib.Geometry.Manifold.PartitionOfUnity\nimport Mathlib.Tactic.FunProp.AEMeasurable\nimport Mathlib.Tactic.FunProp.Measurable\nimport Mathlib.Analysis.Normed.Group.Tannery\nimport Mathlib.Algebra.Order.Field.Basic\nimport Mathlib.Order.Filter.ZeroAndBoundedAtFilter\nimport Mathlib.Analysis.Fourier.RiemannLebesgueLemma\nimport Mathlib.Analysis.SumIntegralComparisons\nimport Mathlib.Algebra.GroupWithZero.Units.Basic\n\nimport PrimeNumberTheoremAnd.BrunTitchmarsh\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence\nimport PrimeNumberTheoremAnd.Fourier\n\n-- note: the opening of ArithmeticFunction introduces a notation \u03c3 that seems\n-- impossible to hide, and hence parameters that are traditionally called \u03c3 will\n-- have to be called \u03c3' instead in this file.\n\nopen Real BigOperators ArithmeticFunction MeasureTheory Filter Set FourierTransform LSeries Asymptotics SchwartzMap\nopen Complex hiding log\nopen scoped Topology\n\nvariable {n : \u2115} {A a b c d u x y t \u03c3' : \u211d} {\u03c8 \u03a8 : \u211d \u2192 \u2102} {F G : \u2102 \u2192 \u2102} {f : \u2115 \u2192 \u2102} {\ud835\udd5c : Type} [RCLike \ud835\udd5c]\n\n-- This version makes the support of \u03a8 explicit, and this is easier for some later proofs\nlemma smooth_urysohn_support_Ioo (h1 : a < b) (h3: c < d) :\n    \u2203 \u03a8 : \u211d \u2192 \u211d, (ContDiff \u211d \u22a4 \u03a8) \u2227 (HasCompactSupport \u03a8) \u2227 Set.indicator (Set.Icc b c) 1 \u2264 \u03a8 \u2227\n    \u03a8 \u2264 Set.indicator (Set.Ioo a d) 1 \u2227 (Function.support \u03a8 = Set.Ioo a d) := by\n\n  have := exists_msmooth_zero_iff_one_iff_of_isClosed\n    (modelWithCornersSelf \u211d \u211d) (s := Set.Iic a \u222a Set.Ici d) (t := Set.Icc b c)\n    (IsClosed.union isClosed_Iic isClosed_Ici)\n    (isClosed_Icc)\n    (by\n      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def, Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc,\n        Set.mem_empty_iff_false, and_imp, imp_false, not_le, Set.mem_Ici]\n      constructor <;> intros <;> linarith)\n\n  rcases this with \u27e8\u03a8, h\u03a8Smooth, h\u03a8range, h\u03a80, h\u03a81\u27e9\n\n  simp only [Set.EqOn, Set.mem_setOf_eq, Set.mem_union, Set.mem_Iic, Set.mem_Ici,\n    ContMDiffMap.coeFn_mk, Pi.zero_apply, Set.mem_Icc, Pi.one_apply, and_imp] at *\n  use \u03a8\n  constructor\n  \u00b7 exact ContMDiff.contDiff h\u03a8Smooth\n  \u00b7 constructor\n    \u00b7 rw [hasCompactSupport_def]\n      apply IsCompact.closure_of_subset (K := Set.Icc a d) isCompact_Icc\n      simp_rw [Function.support_subset_iff, ne_eq, <-h\u03a80]\n      intro x hx\n      contrapose! hx\n      simp only [Set.mem_Icc, not_and_or] at hx\n      by_contra! h'\n      cases' hx <;> linarith\n    \u00b7 constructor\n      \u00b7 intro x\n        rw [Set.indicator_apply]\n        split_ifs with h\n        \u00b7 simp only [Set.mem_Icc, Pi.one_apply] at *\n          simp_rw [h\u03a81 x] at h\n          exact Eq.le (_root_.id h.symm)\n        \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n          have : \u03a8 x \u2208 Set.Icc 0 1 := h\u03a8range this\n          exact this.left\n      \u00b7 constructor\n        \u00b7 intro x\n          rw [Set.indicator_apply]\n          split_ifs with h\n          \u00b7 have : \u03a8 x \u2208 Set.range \u03a8 := by simp only [Set.mem_range, exists_apply_eq_apply]\n            have : \u03a8 x \u2208 Set.Icc 0 1 := by exact h\u03a8range this\n            simpa using this.2\n          \u00b7 simp only [Set.mem_Ioo, Pi.one_apply] at *\n            simp only [not_and_or, not_lt] at h\n            simp_rw [h\u03a80 x] at h\n            exact Eq.le h\n        \u00b7 simp_rw [Function.support, ne_eq, \u2190h\u03a80]\n          push_neg\n          simp [Set.ext_iff]\n  done\n\n\n/-%%\nThe Fourier transform of an absolutely integrable function $\\psi: \\R \\to \\C$ is defined by the formula\n$$ \\hat \\psi(u) := \\int_\\R e(-tu) \\psi(t)\\ dt$$\nwhere $e(\\theta) := e^{2\\pi i \\theta}$.\n\nLet $f: \\N \\to \\C$ be an arithmetic function such that $\\sum_{n=1}^\\infty \\frac{|f(n)|}{n^\\sigma} < \\infty$ for all $\\sigma>1$.  Then the Dirichlet series\n$$ F(s) := \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}$$\nis absolutely convergent for $\\sigma>1$.\n%%-/\n\nnoncomputable\ndef nterm (f : \u2115 \u2192 \u2102) (\u03c3' : \u211d) (n : \u2115) : \u211d := if n = 0 then 0 else \u2016f n\u2016 / n ^ \u03c3'\n\nlemma nterm_eq_norm_term {f : \u2115 \u2192 \u2102} : nterm f \u03c3' n = \u2016term f \u03c3' n\u2016 := by\n  by_cases h : n = 0 <;> simp [nterm, term, h]\n\nlemma hf_coe1 (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (h\u03c3 : 1 < \u03c3') :\n    \u2211' i, (\u2016term f \u03c3' i\u2016\u208a : ENNReal) \u2260 \u22a4 := by\n  simp_rw [ENNReal.tsum_coe_ne_top_iff_summable_coe, \u2190 norm_toNNReal]\n  norm_cast\n  apply Summable.toNNReal\n  convert hf \u03c3' h\u03c3 with i\n  simp [nterm_eq_norm_term]\n\nlemma first_fourier_aux1 (h\u03c8: Continuous \u03c8) {x : \u211d} (n : \u2115) : Measurable fun (u : \u211d) \u21a6\n    (\u2016fourierChar (-(u * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * (n / x).log))) \u2022 \u03c8 u\u2016\u208a : ENNReal) := by\n  -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n  refine Measurable.comp ?_ (by fun_prop) |>.smul (by fun_prop)\n    |>.nnnorm |>.coe_nnreal_ennreal\n  exact Continuous.measurable Real.continuous_fourierChar\n\nlemma first_fourier_aux2a :\n    (2 : \u2102) * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log ((n) / x))) = -(y * ((n) / x).log) := by\n  calc\n    _ = -(y * (((2 : \u2102) * \u03c0) / (2 * \u03c0) * Real.log ((n) / x))) := by ring\n    _ = _ := by rw [div_self (by norm_num; exact pi_ne_zero), one_mul]\n\nlemma first_fourier_aux2 (hx : 0 < x) (n : \u2115) :\n    term f \u03c3' n * \ud835\udc1e (-(y * (1 / (2 * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 y =\n    term f (\u03c3' + y * I) n \u2022 (\u03c8 y * x ^ (y * I)) := by\n  by_cases hn : n = 0 ; simp [term, hn]\n  simp only [term, hn, \u2193reduceIte, fourierChar_apply]\n  calc\n    _ = (f n * (cexp ((2 * \u03c0 * -(y * (1 / (2 * \u03c0) * Real.log (n / x)))) * I) / \u2191((n : \u211d) ^ \u03c3'))) \u2022 \u03c8 y := by\n      have : ((\u2191n : \u2102) ^ (\u03c3' : \u2102) : \u2102) = ((\u2191n : \u211d) ^ (\u03c3' : \u211d) : \u211d) := by\n        rw [Complex.cpow_def_of_ne_zero (by simp [hn]), Real.rpow_def_of_nonneg (Nat.cast_nonneg n)]\n        simp [hn]\n      simp [Real.fourierChar, expMapCircle, smul_eq_mul, mul_assoc, this] ; ring\n    _ = (f n * (x ^ (y * I) / n ^ (\u03c3' + y * I))) \u2022 \u03c8 y := by\n      congr 2\n      have l1 : 0 < (n : \u211d) := by simpa using Nat.pos_iff_ne_zero.mpr hn\n      have l2 : (x : \u2102) \u2260 0 := by simp [hx.ne.symm]\n      have l3 : (n : \u2102) \u2260 0 := by simp [hn]\n      rw [Real.rpow_def_of_pos l1, Complex.cpow_def_of_ne_zero l2, Complex.cpow_def_of_ne_zero l3]\n      push_cast\n      simp_rw [\u2190 Complex.exp_sub]\n      congr 1\n      rw [first_fourier_aux2a, Real.log_div l1.ne.symm hx.ne.symm]\n      push_cast\n      rw [Complex.ofReal_log hx.le]\n      ring\n    _ = _ := by simp ; group\n\n/-%%\n\\begin{lemma}[First Fourier identity]\\label{first-fourier}\\lean{first_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n  $$ \\sum_{n=1}^\\infty \\frac{f(n)}{n^\\sigma} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) = \\int_\\R F(\\sigma + it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\nlemma first_fourier (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (hcont: Continuous \u03c8)\n    (hsupp: Integrable \u03c8) (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u2211' n : \u2115, term f \u03c3' n * (\ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x))) =\n    \u222b t : \u211d, LSeries f (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n/-%%\n\\begin{proof}\\leanok  By the definition of the Fourier transform, the left-hand side expands as\n$$ \\sum_{n=1}^\\infty \\int_\\R \\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x})\\ dt$$\nwhile the right-hand side expands as\n$$ \\int_\\R \\sum_{n=1}^\\infty \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}\\ dt.$$\nSince\n$$\\frac{f(n)}{n^\\sigma} \\psi(t) e( - \\frac{1}{2\\pi} t \\log \\frac{n}{x}) = \\frac{f(n)}{n^{\\sigma+it}} \\psi(t) x^{it}$$\nthe claim then follows from Fubini's theorem.\n\\end{proof}\n%%-/\n  calc\n    _ = \u2211' n, term f \u03c3' n * \u222b (v : \u211d), \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by rfl\n    _ = \u2211' n, \u222b (v : \u211d), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      simp [integral_mul_left]\n    _ = \u222b (v : \u211d), \u2211' (n : \u2115), term f \u03c3' n * \ud835\udc1e (-(v * ((1 : \u211d) / ((2 : \u211d) * \u03c0) * Real.log (n / x)))) \u2022 \u03c8 v := by\n      refine (integral_tsum ?_ ?_).symm\n      \u00b7 -- TODO: attribute [fun_prop] Real.continuous_fourierChar once `fun_prop` bugfix is merged\n        refine fun _ \u21a6 Measurable.aestronglyMeasurable ?_\n        refine Measurable.mul (by fun_prop) ((Measurable.comp ?_ (by fun_prop)).smul (by fun_prop))\n        exact Continuous.measurable Real.continuous_fourierChar\n      \u00b7 simp_rw [nnnorm_mul]\n        push_cast\n        simp_rw [lintegral_const_mul _ (first_fourier_aux1 hcont _)]\n        calc\n          _ = (\u2211' (i : \u2115), (\u2016term f \u03c3' i\u2016\u208a : ENNReal)) * \u222b\u207b (a : \u211d), \u2016\u03c8 a\u2016\u208a \u2202volume := by\n            simp [ENNReal.tsum_mul_right]\n          _ \u2260 \u22a4 := ENNReal.mul_ne_top (hf_coe1 hf h\u03c3)\n            (ne_top_of_lt hsupp.2)\n    _ = _ := by\n      congr 1; ext y\n      simp_rw [mul_assoc (LSeries _ _), \u2190 smul_eq_mul (a := (LSeries _ _)), LSeries]\n      rw [\u2190 tsum_smul_const]\n      \u00b7 congr with n ; exact first_fourier_aux2 hx n\n      \u00b7 apply Summable.of_norm\n        convert hf \u03c3' h\u03c3 with n\n        by_cases h : n = 0\n        \u00b7 simp [nterm, term, h]\n        \u00b7 simp [nterm, term, h]\n          have : (n : \u2102) \u2260 0 := by simp [h]\n          simp [Complex.abs_cpow_of_ne_zero this]\n\n/-%%\n\\begin{lemma}[Second Fourier identity]\\label{second-fourier}\\lean{second_fourier}\\leanok If $\\psi: \\R \\to \\C$ is continuous and compactly supported and $x > 0$, then for any $\\sigma>1$\n$$ \\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} \\hat \\psi(\\frac{u}{2\\pi})\\ du = x^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\n@[continuity]\nlemma continuous_multiplicative_ofAdd : Continuous (\u21d1Multiplicative.ofAdd : \u211d \u2192 \u211d) := \u27e8fun _ \u21a6 id\u27e9\n\nattribute [fun_prop] measurable_coe_nnreal_ennreal\n\nlemma second_fourier_integrable_aux1a (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (x : \u211d) \u21a6 cexp (-((x : \u2102) * ((\u03c3' : \u2102) - 1)))) (Ici (-Real.log x)) := by\n  norm_cast\n  suffices IntegrableOn (fun (x : \u211d) \u21a6 (rexp (-(x * (\u03c3' - 1))))) (Ici (-x.log)) _ from this.ofReal\n  simp_rw [fun (a x : \u211d) \u21a6 (by ring : -(x * a) = -a * x), integrableOn_Ici_iff_integrableOn_Ioi]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_integrable_aux1 (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8) (h\u03c3 : 1 < \u03c3') :\n    let \u03bd : Measure (\u211d \u00d7 \u211d) := (volume.restrict (Ici (-Real.log x))).prod volume\n    Integrable (Function.uncurry fun (u : \u211d) (a : \u211d) \u21a6 ((rexp (-u * (\u03c3' - 1))) : \u2102) \u2022\n    (\ud835\udc1e (Multiplicative.ofAdd (-(a * (u / (2 * \u03c0))))) : \u2102) \u2022 \u03c8 a) \u03bd := by\n  intro \u03bd\n  constructor\n  \u00b7 apply Measurable.aestronglyMeasurable\n    apply MeasureTheory.measurable_uncurry_of_continuous_of_measurable <;> intro i\n    swap; apply Continuous.measurable\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n    \u00b7 apply Continuous.smul\n      \u00b7 continuity\n      \u00b7 apply Continuous.smul\n        \u00b7 apply Continuous.subtype_val\n          simp [Real.fourierChar, expMapCircle, Multiplicative.ofAdd]\n          continuity\n        \u00b7 continuity\n  \u00b7 let f1 : \u211d \u2192 ENNReal := fun a1 \u21a6 \u2191\u2016cexp (-(\u2191a1 * (\u2191\u03c3' - 1)))\u2016\u208a\n    let f2 : \u211d \u2192 ENNReal := fun a2 \u21a6 \u2191\u2016\u03c8 a2\u2016\u208a\n    suffices \u222b\u207b (a : \u211d \u00d7 \u211d), f1 a.1 * f2 a.2 \u2202\u03bd < \u22a4 by simpa [Function.uncurry, HasFiniteIntegral]\n    refine (lintegral_prod_mul ?_ ?_).trans_lt ?_ <;> unfold_let f1 f2; fun_prop; fun_prop\n    exact ENNReal.mul_lt_top (ne_top_of_lt (second_fourier_integrable_aux1a h\u03c3).2)\n      (ne_top_of_lt hsupp.2)\n\nlemma second_fourier_integrable_aux2 (h\u03c3 : 1 < \u03c3') :\n    IntegrableOn (fun (u : \u211d) \u21a6 cexp ((1 - \u2191\u03c3' - \u2191t * I) * \u2191u)) (Ioi (-Real.log x)) := by\n  refine (integrable_norm_iff (Measurable.aestronglyMeasurable <| by fun_prop)).mp ?_\n  suffices IntegrableOn (fun a \u21a6 rexp (-(\u03c3' - 1) * a)) (Ioi (-x.log)) _ by simpa [Complex.abs_exp]\n  apply exp_neg_integrableOn_Ioi\n  linarith\n\nlemma second_fourier_aux (hx : 0 < x) :\n    -(cexp (-((1 - \u2191\u03c3' - \u2191t * I) * \u2191(Real.log x))) / (1 - \u2191\u03c3' - \u2191t * I)) =\n    \u2191(x ^ (\u03c3' - 1)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 * \u2191x ^ (\u2191t * I) := by\n  calc\n    _ = cexp (\u2191(Real.log x) * ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by rw [\u2190 div_neg]; ring_nf\n    _ = (x ^ ((\u2191\u03c3' - 1) + \u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_def_of_ne_zero (ofReal_ne_zero.mpr (ne_of_gt hx)), Complex.ofReal_log hx.le]\n    _ = (x ^ ((\u03c3' : \u2102) - 1)) * (x ^ (\u2191t * I)) * (\u2191\u03c3' + \u2191t * I - 1)\u207b\u00b9 := by\n      rw [Complex.cpow_add _ _ (ofReal_ne_zero.mpr (ne_of_gt hx))]\n    _ = _ := by rw [ofReal_cpow hx.le]; push_cast; ring\n\nlemma second_fourier (hcont: Continuous \u03c8) (hsupp: Integrable \u03c8)\n    {x \u03c3' : \u211d} (hx : 0 < x) (h\u03c3 : 1 < \u03c3') :\n    \u222b u in Ici (-log x), Real.exp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    (x^(\u03c3' - 1) : \u211d) * \u222b t, (1 / (\u03c3' + t * I - 1)) * \u03c8 t * x^(t * I) \u2202 volume := by\n/-%%\n\\begin{proof}\\leanok\nThe left-hand side expands as\n$$ \\int_{-\\log x}^\\infty \\int_\\R e^{-u(\\sigma-1)} \\psi(t) e(-\\frac{tu}{2\\pi})\\ dt\\ du =\nx^{\\sigma - 1} \\int_\\R \\frac{1}{\\sigma+it-1} \\psi(t) x^{it}\\ dt$$\nso by Fubini's theorem it suffices to verify the identity\n\\begin{align*}\n\\int_{-\\log x}^\\infty e^{-u(\\sigma-1)} e(-\\frac{tu}{2\\pi})\\ du\n&= \\int_{-\\log x}^\\infty e^{(it - \\sigma + 1)u}\\ du \\\\\n&= \\frac{1}{it - \\sigma + 1} e^{(it - \\sigma + 1)u}\\ \\Big|_{-\\log x}^\\infty \\\\\n&= x^{\\sigma - 1} \\frac{1}{\\sigma+it-1} x^{it}\n\\end{align*}\n\\end{proof}\n%%-/\n  conv in \u2191(rexp _) * _ => { rw [Real.fourierIntegral_real_eq, \u2190 smul_eq_mul, \u2190 integral_smul] }\n  rw [MeasureTheory.integral_integral_swap] ; swap ; exact second_fourier_integrable_aux1 hcont hsupp h\u03c3\n  rw [\u2190 integral_mul_left]\n  congr 1; ext t\n  dsimp [Real.fourierChar, expMapCircle]\n  simp_rw [\u2190 mul_assoc _ _ (\u03c8 _), integral_mul_right]\n  rw [fun (a b d : \u2102) \u21a6 show a * (b * (\u03c8 t) * d) = (a * b * d) * \u03c8 t by ring]\n  congr 1\n  push_cast\n  simp_rw [\u2190 Complex.exp_add]\n  have (u : \u211d) :\n      -\u2191u * (\u2191\u03c3' - 1) + 2 * \u2191\u03c0 * -(\u2191t * (\u2191u / (2 * \u2191\u03c0))) * I = (1 - \u03c3' - t * I) * u := calc\n    _ = -\u2191u * (\u2191\u03c3' - 1) + (2 * \u2191\u03c0) / (2 * \u2191\u03c0) * -(\u2191t * \u2191u) * I := by ring\n    _ = -\u2191u * (\u2191\u03c3' - 1) + 1 * -(\u2191t * \u2191u) * I := by rw [div_self (by norm_num; exact pi_ne_zero)]\n    _ = _ := by ring\n  simp_rw [this]\n  let c : \u2102 := (1 - \u2191\u03c3' - \u2191t * I)\n  have : c \u2260 0 := by simp [Complex.ext_iff, c] ; intro h ; linarith\n  let f' (u : \u211d) := cexp (c * u)\n  let f := fun (u : \u211d) \u21a6 (f' u) / c\n  have hderiv : \u2200 u \u2208 Ici (-Real.log x), HasDerivAt f (f' u) u := by\n    intro u _\n    rw [show f' u = cexp (c * u) * (c * 1) / c by field_simp]\n    exact (hasDerivAt_id' u).ofReal_comp.const_mul c |>.cexp.div_const c\n  have hf : Tendsto f atTop (\ud835\udcdd 0) := by\n    apply tendsto_zero_iff_norm_tendsto_zero.mpr\n    suffices Tendsto (fun (x : \u211d) \u21a6 abs (cexp (c * \u2191x)) / abs c) atTop (\ud835\udcdd (0 / abs c)) by simpa [f, f'] using this\n    apply Filter.Tendsto.div_const\n    suffices Tendsto (. * (1 - \u03c3')) atTop atBot by simpa [Complex.abs_exp, mul_comm (1 - \u03c3'), c]\n    exact Tendsto.atTop_mul_neg_const (by linarith) fun \u2983s\u2984 h \u21a6 h\n  rw [integral_Ici_eq_integral_Ioi,\n    integral_Ioi_of_hasDerivAt_of_tendsto' hderiv (second_fourier_integrable_aux2 h\u03c3) hf]\n  simpa [f, f'] using second_fourier_aux hx\n\n/-%%\nNow let $A \\in \\C$, and suppose that there is a continuous function $G(s)$ defined on $\\mathrm{Re} s \\geq 1$ such that $G(s) = F(s) - \\frac{A}{s-1}$ whenever $\\mathrm{Re} s > 1$.  We also make the Chebyshev-type hypothesis\n\\begin{equation}\\label{cheby}\n\\sum_{n \\leq x} |f(n)| \\ll x\n\\end{equation}\nfor all $x \\geq 1$ (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).\n%%-/\n\nlemma one_add_sq_pos (u : \u211d) : 0 < 1 + u ^ 2 := zero_lt_one.trans_le (by simpa using sq_nonneg u)\n\n/-%%\n\\begin{lemma}[Decay bounds]\\label{decay}\\lean{decay_bounds}\\leanok  If $\\psi:\\R \\to \\C$ is $C^2$ and obeys the bounds\n  $$ |\\psi(t)|, |\\psi''(t)| \\leq A / (1 + |t|^2)$$\n  for all $t \\in \\R$, then\n$$ |\\hat \\psi(u)| \\leq C A / (1+|u|^2)$$\nfor all $u \\in \\R$, where $C$ is an absolute constant.\n\\end{lemma}\n%%-/\n\nlemma decay_bounds_key (f : W21) (u : \u211d) : \u2016\ud835\udcd5 f u\u2016 \u2264 \u2016f\u2016 * (1 + u ^ 2)\u207b\u00b9 := by\n  have l1 : 0 < 1 + u ^ 2 := one_add_sq_pos _\n  have l2 : 1 + u ^ 2 = \u2016(1 : \u2102) + u ^ 2\u2016 := by\n    norm_cast ; simp only [Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.2 l1.le]\n  have l3 : \u20161 / ((4 : \u2102) * \u2191\u03c0 ^ 2)\u2016 \u2264 (4 * \u03c0 ^ 2)\u207b\u00b9 := by simp\n  have key := fourierIntegral_self_add_deriv_deriv f u\n  simp only [Function.iterate_succ _ 1, Function.iterate_one, Function.comp_apply] at key\n  rw [F_sub f.hf (f.hf''.const_mul (1 / (4 * \u2191\u03c0 ^ 2)))] at key\n  rw [\u2190 div_eq_mul_inv, le_div_iff l1, mul_comm, l2, \u2190 norm_mul, key, sub_eq_add_neg]\n  apply norm_add_le _ _ |>.trans\n  change _ \u2264 W21.norm _\n  rw [norm_neg, F_mul, norm_mul, W21.norm]\n  gcongr <;> apply VectorFourier.norm_fourierIntegral_le_integral_norm\n\nlemma decay_bounds_aux {f : \u211d \u2192 \u2102} (hf : AEStronglyMeasurable f volume) (h : \u2200 t, \u2016f t\u2016 \u2264 A * (1 + t ^ 2)\u207b\u00b9) :\n    \u222b t, \u2016f t\u2016 \u2264 \u03c0 * A := by\n  have l1 : Integrable (fun x \u21a6 A * (1 + x ^ 2)\u207b\u00b9) := integrable_inv_one_add_sq.const_mul A\n  simp_rw [\u2190 integral_univ_inv_one_add_sq, mul_comm, \u2190 integral_mul_left]\n  exact integral_mono (l1.mono' hf (eventually_of_forall h)).norm l1 h\n\ntheorem decay_bounds_W21 (f : W21) (hA : \u2200 t, \u2016f t\u2016 \u2264 A / (1 + t ^ 2))\n    (hA' : \u2200 t, \u2016deriv (deriv f) t\u2016 \u2264 A / (1 + t ^ 2)) (u) :\n    \u2016\ud835\udcd5 f u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  have l0 : 1 * (4 * \u03c0)\u207b\u00b9 * A = (4 * \u03c0 ^ 2)\u207b\u00b9 * (\u03c0 * A) := by field_simp ; ring\n  have l1 : \u222b (v : \u211d), \u2016f v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA\n  have l2 : \u222b (v : \u211d), \u2016deriv (deriv f) v\u2016 \u2264 \u03c0 * A := by\n    apply decay_bounds_aux f.deriv.deriv.continuous.aestronglyMeasurable\n    simp_rw [\u2190 div_eq_mul_inv] ; exact hA'\n  apply decay_bounds_key f u |>.trans\n  change W21.norm _ * _ \u2264 _\n  simp_rw [W21.norm, div_eq_mul_inv, add_mul, l0] ; gcongr\n\nlemma decay_bounds (\u03c8 : CS 2 \u2102) (hA : \u2200 t, \u2016\u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) (hA' : \u2200 t, \u2016deriv^[2] \u03c8 t\u2016 \u2264 A / (1 + t ^ 2)) :\n    \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 (\u03c0 + 1 / (4 * \u03c0)) * A / (1 + u ^ 2) := by\n  exact decay_bounds_W21 \u03c8 hA hA' u\n\nlemma decay_bounds_cor_aux (\u03c8 : CS 2 \u2102) : \u2203 C : \u211d, \u2200 u, \u2016\u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  have l1 : HasCompactSupport (fun u : \u211d => ((1 + u ^ 2) : \u211d) * \u03c8 u) := by exact \u03c8.h2.mul_left\n  have := \u03c8.h1.continuous\n  obtain \u27e8C, hC\u27e9 := l1.exists_bound_of_continuous (by continuity)\n  refine \u27e8C, fun u => ?_\u27e9\n  specialize hC u\n  simp only [norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_eq_self.mpr (one_add_sq_pos u).le] at hC\n  rwa [le_div_iff' (one_add_sq_pos _)]\n\nlemma decay_bounds_cor (\u03c8 : W21) :\n    \u2203 C : \u211d, \u2200 u, \u2016\ud835\udcd5 \u03c8 u\u2016 \u2264 C / (1 + u ^ 2) := by\n  simpa only [div_eq_mul_inv] using \u27e8_, decay_bounds_key \u03c8\u27e9\n\n@[continuity] lemma continuous_FourierIntegral (\u03c8 : W21) : Continuous (\ud835\udcd5 \u03c8) :=\n  VectorFourier.fourierIntegral_continuous continuous_fourierChar (by exact continuous_mul) \u03c8.hf\n\nlemma W21.integrable_fourier (\u03c8 : W21) (hc : c \u2260 0) :\n    Integrable fun u \u21a6 \ud835\udcd5 \u03c8 (u / c) := by\n  have l1 (C) : Integrable (fun u \u21a6 C / (1 + (u / c) ^ 2)) volume := by\n    simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C\n  have l2 : AEStronglyMeasurable (fun u \u21a6 \ud835\udcd5 \u03c8 (u / c)) volume := by\n    apply Continuous.aestronglyMeasurable ; continuity\n  obtain \u27e8C, h\u27e9 := decay_bounds_cor \u03c8\n  apply @Integrable.mono' \u211d \u2102 _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_\n  apply eventually_of_forall (fun x => h _)\n\n/-%%\n\\begin{proof} \\leanok From two integration by parts we obtain the identity\n$$ (1+u^2) \\hat \\psi(u) = \\int_{\\bf R} (\\psi(t) - \\frac{u}{4\\pi^2} \\psi''(t)) e(-tu)\\ dt.$$\nNow apply the triangle inequality and the identity $\\int_{\\bf R} \\frac{dt}{1+t^2}\\ dt = \\pi$ to obtain the claim with $C = \\pi + 1 / 4 \\pi$.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[Limiting Fourier identity]\\label{limiting}\\lean{limiting_fourier}\\leanok  If $\\psi: \\R \\to \\C$ is $C^2$ and compactly supported and $x \\geq 1$, then\n$$ \\sum_{n=1}^\\infty \\frac{f(n)}{n} \\hat \\psi( \\frac{1}{2\\pi} \\log \\frac{n}{x} ) - A \\int_{-\\log x}^\\infty \\hat \\psi(\\frac{u}{2\\pi})\\ du =  \\int_\\R G(1+it) \\psi(t) x^{it}\\ dt.$$\n\\end{lemma}\n%%-/\n\nlemma continuous_LSeries_aux (hf : Summable (nterm f \u03c3')) :\n    Continuous fun x : \u211d => LSeries f (\u03c3' + x * I) := by\n\n  have l1 i : Continuous fun x : \u211d \u21a6 term f (\u03c3' + x * I) i := by\n    by_cases h : i = 0\n    \u00b7 simpa [h] using continuous_const\n    \u00b7 simpa [h] using continuous_const.div (continuous_const.cpow (by continuity) (by simp [h])) (fun x => by simp [h])\n  have l2 n (x : \u211d) : \u2016term f (\u03c3' + x * I) n\u2016 = nterm f \u03c3' n := by\n    by_cases h : n = 0\n    \u00b7 simp [h, nterm]\n    \u00b7 field_simp [h, nterm, cpow_add _ _ (Nat.cast_ne_zero.mpr h)]\n      rw [\u2190 Complex.norm_eq_abs, Complex.norm_natCast_cpow_of_pos (Nat.pos_of_ne_zero h)]\n      simp\n  exact continuous_tsum l1 hf (fun n x => le_of_eq (l2 n x))\n\n-- Here compact support is used but perhaps it is not necessary\nlemma limiting_fourier_aux (hG' : Set.EqOn G (fun s \u21a6 LSeries f s - A / (s - 1)) {s | 1 < s.re})\n    (hf : \u2200 (\u03c3' : \u211d), 1 < \u03c3' \u2192 Summable (nterm f \u03c3')) (\u03c8 : CS 2 \u2102) (hx : 1 \u2264 x) (\u03c3' : \u211d) (h\u03c3' : 1 < \u03c3') :\n    \u2211' n, term f \u03c3' n * \ud835\udcd5 \u03c8 (1 / (2 * \u03c0) * log (n / x)) -\n    A * (x ^ (1 - \u03c3') : \u211d) * \u222b u in Ici (- log x), rexp (-u * (\u03c3' - 1)) * \ud835\udcd5 \u03c8 (u / (2 * \u03c0)) =\n    \u222b t : \u211d, G (\u03c3' + t * I) * \u03c8 t * x ^ (t * I) := by\n\n  have hint : Integrable \u03c8 := \u03c8.h1.continuous.integrable_of_hasCompactSupport \u03c8.h2\n  have l3 : 0 < x := zero_lt_one.trans_le hx\n  have l1 (\u03c3') (h\u03c3' : 1 < \u03c3') := first_fourier hf \u03c8.h1.continuous hint l3 h\u03c3'\n  have l2 (\u03c3') (h\u03c3' : 1 < \u03c3') := second_fourier \u03c8.h1.continuous hint l3 h\u03c3'\n  have l8 : Continuous fun t : \u211d \u21a6 (x : \u2102) ^ (t * I) :=\n    continuous_const.cpow (continuous_ofReal.mul continuous_const) (by simp [l3])\n  have l6 : Continuous fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    apply ((continuous_LSeries_aux (hf _ h\u03c3')).mul \u03c8.h1.continuous).mul l8\n  have l4 : Integrable fun t : \u211d \u21a6 LSeries f (\u2191\u03c3' + \u2191t * I) * \u03c8 t * \u2191x ^ (\u2191t * I) := by\n    exact l6.integrable_of_hasCompactSupport \u03c8.h2.mul_left.mul_right\n  have e2 (u : \u211d) : \u03c3' + u * I - 1 \u2260 0 := by\n    intro h ; have := congr_arg Complex.re h ; simp at this ; linarith\n  have l7 : Continuous fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    simp [\u2190 mul_assoc]\n    refine ((continuous_const.mul <| Continuous.inv\u2080 ?_ e2).mul \u03c8.h1.continuous).mul l8\n    continuity\n  have l5 : Integrable fun a \u21a6 A * \u2191(x ^ (1 - \u03c3')) * (\u2191(x ^ (\u03c3' - 1)) * (1 / (\u03c3' + a * I - 1) * \u03c8 a * x ^ (a * I))) := by\n    apply l7.integrable_of_hasCompactSupport\n    exact \u03c8.h2.mul_left.mul_right.mul_left.mul_left\n\n  simp_rw [l1 \u03c3' h\u03c3', l2 \u03c3' h\u03c3', \u2190 integral_mul_left, \u2190 integral_sub l4 l5]\n  apply integral_congr_ae\n  apply eventually_of_forall\n  intro u\n  have e1 : 1 < ((\u03c3' : \u2102) + (u : \u2102) * I).re := by simp [h\u03c3']\n  simp_rw [hG' e1, sub_mul, \u2190 mul_assoc]\n  field_simp [e2] ; left ; left\n  norm_cast\n  simp [mul_assoc, \u2190 rpow_add l3]\n\nsection nabla\n\nvariable {\u03b1 E : Type*} [OfNat \u03b1 1] [Add \u03b1] [Sub \u03b1] {u : \u03b1 \u2192 \u2102}\n\ndef cumsum [AddCommMonoid E] (u : \u2115 \u2192 E) (n : \u2115) : E := \u2211 i in Finset.range n, u i\n\ndef nabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1) - u n\n\n/- TODO nnabla is redundant -/\ndef nnabla [Sub E] (u : \u03b1 \u2192 E) (n : \u03b1) : E := u n - u (n + 1)\n\ndef shift (u : \u03b1 \u2192 E) (n : \u03b1) : E := u (n + 1)\n\n@[simp] lemma cumsum_zero [AddCommMonoid E] {u : \u2115 \u2192 E} : cumsum u 0 = 0 := by simp [cumsum]\n\nlemma cumsum_succ [AddCommMonoid E] {u : \u2115 \u2192 E} (n : \u2115) :\n    cumsum u (n + 1) = cumsum u n + u n := by\n  simp [cumsum, Finset.sum_range_succ]\n\n@[simp] lemma nabla_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : nabla (cumsum u) = u := by\n  ext n ; simp [nabla, cumsum, Finset.range_succ]\n\nlemma neg_cumsum [AddCommGroup E] {u : \u2115 \u2192 E} : -(cumsum u) = cumsum (-u) := funext (fun n => by simp [cumsum])\n\nlemma cumsum_nonneg {u : \u2115 \u2192 \u211d} (hu : 0 \u2264 u) : 0 \u2264 cumsum u := fun _ => Finset.sum_nonneg (fun i _ => hu i)\n\n", "theoremStatement": "lemma neg_nabla [Ring E] {u : \u03b1 \u2192 E} : -(nabla u) = nnabla u ", "theoremName": "neg_nabla", "fileCreated": {"commit": "a4d6b1db8f54183d1584340c91021ade2768f8e4", "date": "2024-01-23"}, "theoremCreated": {"commit": "cb638ee83f4871252e8d084142297b3124619e5c", "date": "2024-03-09"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Wiener.lean", "module": "PrimeNumberTheoremAnd.Wiener", "jsonFile": "PrimeNumberTheoremAnd.Wiener.jsonl", "positionMetadata": {"lineInFile": 495, "tokenPositionInFile": 23893, "theoremPositionInFile": 34}, "dependencyMetadata": 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"Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Star", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.Data.Real.EReal", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Analysis.Analytic.Basic", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.Analysis.Convolution", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", 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"Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier"]}, "proofMetadata": {"hasProof": true, "proof": ":= by ext n ; simp [nabla, nnabla]", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 34}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.NumberTheory.PrimeCounting\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Data.Complex.ExponentialBounds\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\n/-!\n# Bounds for the Selberg sieve\nThis file proves a number of results to help bound `Sieve.selbergSum`\n\n## Main Results\n* `selbergBoundingSum_ge_sum_div`: If `\u03bd` is completely multiplicative then `S \u2265 \u2211_{n \u2264 \u221ay}, \u03bd n`\n* `boundingSum_ge_log`: If `\u03bd n = 1 / n` then `S \u2265 log y / 2`\n* `rem_sum_le_of_const`: If `R_d \u2264 C` then the error term is at most `C * y * (1 + log y)^3`\n-/\n\nopen scoped Nat ArithmeticFunction BigOperators Classical\n\nnoncomputable section\nnamespace Sieve\n\nlemma prodDistinctPrimes_squarefree (s : Finset \u2115) (h : \u2200 p \u2208 s, p.Prime) :\n    Squarefree (\u220f p in s, p) := by\n  refine Iff.mpr Nat.squarefree_iff_prime_squarefree ?_\n  intro p hp; by_contra h_dvd\n  by_cases hps : p \u2208 s\n  \u00b7 rw [\u2190Finset.mul_prod_erase (a:=p) (h := hps), mul_dvd_mul_iff_left (Nat.Prime.ne_zero hp)] at h_dvd\n    cases' Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) h_dvd with q hq\n    rw [Finset.mem_erase] at hq\n    exact hq.1.1 $ symm $ (Nat.prime_dvd_prime_iff_eq hp (h q hq.1.2)).mp hq.2\n  \u00b7 have : p \u2223 \u220f p in s, p := Trans.trans (dvd_mul_right p p) h_dvd\n    cases' Prime.exists_mem_finset_dvd (Nat.Prime.prime hp) this with q hq\n    have heq : p = q := by\n      rw [\u2190Nat.prime_dvd_prime_iff_eq hp (h q hq.1)]\n      exact hq.2\n    rw [heq] at hps; exact hps hq.1\n\nlemma primorial_squarefree (n : \u2115) : Squarefree (primorial n) := by\n  apply prodDistinctPrimes_squarefree\n  simp_rw [Finset.mem_filter];\n  exact fun _ h => h.2\n\ntheorem zeta_pos_of_prime : \u2200 (p : \u2115), Nat.Prime p \u2192 (0:\u211d) < (\u2191\u03b6:ArithmeticFunction \u211d) p := by\n  intro p hp\n  rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, if_neg (Nat.Prime.ne_zero hp)]\n  norm_num\n\n", "theoremStatement": "theorem zeta_lt_self_of_prime : \u2200 (p : \u2115), Nat.Prime p \u2192 (\u2191\u03b6:ArithmeticFunction \u211d) p < (p:\u211d) ", "theoremName": "Sieve.zeta_lt_self_of_prime", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/SelbergBounds.lean", "module": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds.jsonl", "positionMetadata": {"lineInFile": 54, "tokenPositionInFile": 2100, "theoremPositionInFile": 3}, 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"Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Data.Nat.Bits", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Prime", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Int.ModEq", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Tactic.Abel", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Order", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  intro p hp\n  rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, if_neg (Nat.Prime.ne_zero hp)]\n  norm_num;\n  exact Nat.succ_le.mp (Nat.Prime.two_le hp)", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 176}}
+{"srcContext": "/-\nCopyright (c) 2024 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Analysis.Asymptotics.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics\nimport Mathlib.Analysis.SpecialFunctions.Log.Base\nimport Mathlib.NumberTheory.Primorial\nimport Mathlib.Data.Complex.ExponentialBounds\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg\nimport PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds\n\nopen Sieve\nopen Filter Asymptotics\nopen scoped Nat ArithmeticFunction BigOperators\n\nnoncomputable section\nnamespace BrunTitchmarsh\n\n/- Sifting primes \u2264 z from the interval [x, x+y] -/\ndef primeInterSieve (x y z : \u211d) (hz : 1 \u2264 z): SelbergSieve := {\n  support := Finset.Icc (Nat.ceil x) (Nat.floor (x+y))\n  prodPrimes := primorial (Nat.floor z)\n  prodPrimes_squarefree := primorial_squarefree _\n  weights := fun _ => 1\n  weights_nonneg := fun _ => zero_le_one\n  totalMass := y\n  nu := (\u03b6 : ArithmeticFunction \u211d).pdiv .id\n  nu_mult := by arith_mult\n  nu_pos_of_prime := fun p hp _ => by\n    simp[if_neg hp.ne_zero, Nat.pos_of_ne_zero hp.ne_zero]\n  nu_lt_one_of_prime := fun p hp _ => by\n    simp[hp.ne_zero]\n    apply inv_lt_one\n    norm_cast\n    exact hp.one_lt\n  level := z\n  one_le_level := hz\n}\n\n/- The number of primes in the interval [a, b] -/\ndef primesBetween (a b : \u211d) : \u2115 :=\n  (Finset.Icc (Nat.ceil a) (Nat.floor b)).filter (Nat.Prime) |>.card\n\nvariable (x y z : \u211d) (hx : 0 < x) (hy : 0 < y) (hz : 1 \u2264 z)\n\nopen Classical in\ntheorem siftedSum_eq_card :\n    (primeInterSieve x y z hz).siftedSum =\n      ((Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d)).card := by\n  apply Sieve.siftedSum_eq\n  exact fun _ _ => rfl\n  exact hz\n  rfl\n\nopen Classical in\n", "theoremStatement": "theorem primesBetween_subset :\n  (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (Nat.Prime) \u2286\n    (Finset.Icc (Nat.ceil x) (Nat.floor (x+y))).filter (fun d => \u2200 p:\u2115, p.Prime \u2192 p \u2264 z \u2192 \u00acp \u2223 d) \u222a\n    (Finset.Icc 1 (Nat.floor z)) ", "theoremName": "BrunTitchmarsh.primesBetween_subset", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/BrunTitchmarsh.lean", "module": "PrimeNumberTheoremAnd.BrunTitchmarsh", "jsonFile": "PrimeNumberTheoremAnd.BrunTitchmarsh.jsonl", "positionMetadata": {"lineInFile": 60, "tokenPositionInFile": 1903, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 228, "importedModules": 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"Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.Analysis.SumIntegralComparisons", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  intro p\n  simp\n  intro hx hxy hp\n  by_cases hpz : p \u2264 z\n  \u00b7 right\n    rw[Nat.le_floor_iff (by linarith)]\n    have := hp.ne_zero\n    exact \u27e8by omega, hpz\u27e9\n  \u00b7 left\n    refine \u27e8\u27e8hx, hxy\u27e9, ?_\u27e9\n    intro q hq hqz\n    rw[hp.dvd_iff_eq (hq.ne_one)]\n    rintro rfl\n    exact hpz hqz", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 283}}
+{"srcContext": "/-\nCopyright (c) 2024 Lawrence Wu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Lawrence Wu\n-/\n\nimport Mathlib.MeasureTheory.Integral.SetIntegral\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta\n\n/-!\n# Uniform Asymptotics\n\nFor a family of functions `f : \u03b9 \u00d7 \u03b1 \u2192 E` and `g : \u03b1 \u2192 E`, we can think of\n`f =O[\ud835\udcdf s \u00d7\u02e2 l] fun (i, x) \u21a6 g x` as expressing that `f i` is O(g) uniformly on `s`.\n\nThis file provides methods for constructing `=O[\ud835\udcdf s \u00d7\u02e2 l]` relations (similarly `\u0398`)\nand deriving their consequences.\n-/\n\nopen Filter\n\nopen Topology\n\nnamespace Asymptotics\n\nvariable {\u03b1 \u03b9 E F : Type*} {s : Set \u03b9}\n\nsection Basic\n\nvariable [Norm E] [Norm F] {f : \u03b9 \u00d7 \u03b1 \u2192 E} {g : \u03b1 \u2192 F} {l : Filter \u03b1}\n\n/-- If f = O(g) uniformly on `s`, then f_i = O(g) for any i.` -/\ntheorem isBigO_of_isBigOUniformly (h : f =O[\ud835\udcdf s \u00d7\u02e2 l] (g \u2218 Prod.snd)) {i : \u03b9} (hi : i \u2208 s) :\n    (fun x \u21a6 f (i, x)) =O[l] g := by\n  obtain \u27e8C, hC\u27e9 := h.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  refine isBigO_iff.mpr \u27e8C, Filter.eventually_iff_exists_mem.mpr \u27e8v, hv, ?_\u27e9\u27e9\n  exact fun y hy \u21a6 ht _ <| huv \u27e8hu hi, hy\u27e9\n\n/-- If f = \u03a9(g) uniformly on `s`, then f_i = \u03a9(g) for any i.` -/\ntheorem isBigO_rev_of_isBigOUniformly_rev (h : (g \u2218 Prod.snd) =O[\ud835\udcdf s \u00d7\u02e2 l] f) {i : \u03b9} (hi : i \u2208 s) :\n    g =O[l] fun x \u21a6 f (i, x) := by\n  obtain \u27e8C, hC\u27e9 := h.bound\n  obtain \u27e8t, htl, ht\u27e9 := hC.exists_mem\n  obtain \u27e8u, hu, v, hv, huv\u27e9 := Filter.mem_prod_iff.mp htl\n  refine isBigO_iff.mpr \u27e8C, Filter.eventually_iff_exists_mem.mpr \u27e8v, hv, ?_\u27e9\u27e9\n  exact fun y hy \u21a6 ht (i, y) <| huv \u27e8hu hi, hy\u27e9\n\n", "theoremStatement": "/-- If f = \u0398(g) uniformly on `s`, then f_i = \u0398(g) for any i.` -/\ntheorem isTheta_of_isThetaUniformly (h : f =\u0398[\ud835\udcdf s \u00d7\u02e2 l] (g \u2218 Prod.snd)) {i : \u03b9} (hi : i \u2208 s) :\n    (fun x \u21a6 f (i, x)) =\u0398[l] g ", "theoremName": "Asymptotics.isTheta_of_isThetaUniformly", "fileCreated": {"commit": "84fbebeb60a5e72e2d8a4e6b350a46af7023f681", "date": "2024-02-08"}, "theoremCreated": {"commit": "84fbebeb60a5e72e2d8a4e6b350a46af7023f681", "date": "2024-02-08"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Analysis/Asymptotics/Uniformly.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly.jsonl", "positionMetadata": {"lineInFile": 50, "tokenPositionInFile": 1648, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, 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"Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  \u27e8isBigO_of_isBigOUniformly h.1 hi, isBigO_rev_of_isBigOUniformly_rev h.2 hi\u27e9", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 81}}
+{"srcContext": "/-\nCopyright (c) 2023 Arend Mellendijk. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Arend Mellendijk\n-/\n\nimport Mathlib.Logic.Embedding.Basic\nimport Mathlib.NumberTheory.ArithmeticFunction\n\n\n/-!\n# Sets of tuples with a fixed product\nThis file defines the finite set of `d`-tuples of natural numbers with a fixed product `n` as\n`Nat.finMulAntidiagonal`.\n## Main Results\n* There are `d^(\u03c9 n)` ways to write `n` as a product of `d` natural numbers, when `n` is squarefree\n(`card_finMulAntidiagonal_of_squarefree`)\n* There are `3^(\u03c9 n)` pairs of natural numbers whose `lcm` is `n`, when `n` is squarefree\n(`card_pair_lcm_eq`)\n\nSee PR #10668\n-/\n\nopen Finset\nopen scoped BigOperators ArithmeticFunction\n\n\nnamespace Nat\ndef finMulAntidiagonal (d : \u2115) (n : \u2115) : Finset (Fin d \u2192 \u2115) :=\n  aux d n\nwhere\n  /-- Auxiliary construction for `finMulAntidiagonal` that bundles a proof of lawfulness\n  (`mem_finMulAntidiagonal`), as this is needed to invoke `disjiUnion`. Using `Finset.disjiUnion` makes\n  this computationally much more efficient than using `Finset.biUnion`. -/\n  aux (d : \u2115) (n : \u2115) : {s : Finset (Fin d \u2192 \u2115) // \u2200 f, f \u2208 s \u2194 \u220f i, f i = n \u2227 n \u2260 0} :=\n    match d with\n    | 0 =>\n      if h : n = 1 then\n        \u27e8{1}, by simp [h]; exact List.ofFn_inj.mp rfl\u27e9\n      else\n        \u27e8\u2205, by simp [Ne.symm h]\u27e9\n    | d + 1 =>\n      { val := (divisorsAntidiagonal n).disjiUnion\n          (fun ab => (aux d ab.2).1.map {\n              toFun := Fin.cons (ab.1)\n              inj' := Fin.cons_right_injective _ })\n          (fun i _hi j _hj hij => Finset.disjoint_left.2 fun t hti htj => hij <| by\n            simp_rw [Finset.mem_map, Function.Embedding.coeFn_mk] at hti htj\n            obtain \u27e8ai, hai, hij'\u27e9 := hti\n            obtain \u27e8aj, haj, rfl\u27e9 := htj\n            rw [Fin.cons_eq_cons] at hij'\n            ext\n            \u00b7 exact hij'.1\n            \u00b7 obtain \u27e8-, rfl\u27e9 := hij'\n              rw [\u2190 (aux d i.2).prop ai |>.mp hai |>.1, \u2190 (aux d j.2).prop ai |>.mp haj |>.1])\n        property := fun f => by\n          simp_rw [mem_disjiUnion, mem_divisorsAntidiagonal, mem_map, Function.Embedding.coeFn_mk,\n            Prod.exists, (aux d _).prop, Fin.prod_univ_succ]\n          constructor\n          \u00b7 rintro \u27e8a, b, \u27e8rfl, hab\u27e9, g, \u27e8rfl, hb\u27e9, rfl\u27e9\n            simp only [Fin.cons_zero, Fin.cons_succ]\n            exact (true_and_iff (a * \u220f x : Fin d, g x \u2260 0)).mpr hab\n          \u00b7 intro \u27e8rfl, hf\u27e9\n            exact \u27e8_, _, \u27e8rfl, hf\u27e9, _, \u27e8rfl, by exact right_ne_zero_of_mul hf\u27e9, Fin.cons_self_tail f\u27e9 }\n\n@[simp]\ntheorem mem_finMulAntidiagonal {d n : \u2115} {f : (Fin d) \u2192 \u2115} :\n    f \u2208 finMulAntidiagonal d n \u2194 \u220f i, f i = n \u2227 n \u2260 0 :=\n  (finMulAntidiagonal.aux d n).prop f\n\n@[simp]\ntheorem finMulAntidiagonal_zero {d : \u2115} :\n    finMulAntidiagonal d 0 = \u2205 := by\n  ext; simp\n\ntheorem finMulAntidiagonal_one {d : \u2115} :\n    finMulAntidiagonal d 1 = {fun _ => 1} := by\n  ext f; simp only [mem_finMulAntidiagonal, and_true, mem_singleton]\n  constructor\n  \u00b7 intro \u27e8hf, _\u27e9; ext i;\n    rw [\u2190Nat.dvd_one, \u2190hf];\n    exact dvd_prod_of_mem f (mem_univ _)\n  \u00b7 rintro rfl; simp only [prod_const_one, implies_true, ne_eq, one_ne_zero, not_false_eq_true,\n    and_self]\n\ntheorem finMulAntidiagonal_empty_of_ne_one {n : \u2115} (hn : n \u2260 1) :\n    finMulAntidiagonal 0 n = \u2205 := by\n  ext; simp [hn.symm]\n\ntheorem dvd_of_mem_finMulAntidiagonal {n d : \u2115} {f : Fin d \u2192 \u2115} (hf : f \u2208 finMulAntidiagonal d n) (i : Fin d):\n    f i \u2223 n := by\n  rw [mem_finMulAntidiagonal] at hf\n  rw [\u2190hf.1]\n  exact dvd_prod_of_mem f (mem_univ i)\n\n", "theoremStatement": "theorem ne_zero_of_mem_finMulAntidiagonal {d n : \u2115} {f : Fin d \u2192 \u2115} (hf : f \u2208 finMulAntidiagonal d n) (i : Fin d) :\n    f i \u2260 0 ", "theoremName": "Nat.ne_zero_of_mem_finMulAntidiagonal", "fileCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "theoremCreated": {"commit": "9b93876e4a090d14832da88d2328036df74035bc", "date": "2024-04-01"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/Mathlib/Data/Nat/FinMulAntidiagonal.lean", "module": "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "jsonFile": "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal.jsonl", "positionMetadata": {"lineInFile": 96, "tokenPositionInFile": 3526, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 22, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", 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"Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.NumberTheory.Divisors", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Data.Rat.BigOperators", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Data.Finsupp.Basic", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.Closure", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Tactic.FinCases", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.GroupTheory.Finiteness", "Mathlib.RingTheory.Finiteness", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.Multiplicity", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.PrimeFin", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.Data.Nat.Multiplicity", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  ne_zero_of_dvd_ne_zero (mem_finMulAntidiagonal.mp hf).2 (dvd_of_mem_finMulAntidiagonal hf i)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 97}}
+{"srcContext": "import Mathlib.Analysis.Calculus.ContDiff.Basic\nimport PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly\nimport PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics\nimport PrimeNumberTheoremAnd.ResidueCalcOnRectangles\nimport PrimeNumberTheoremAnd.Wiener\n\nopen Asymptotics Complex ComplexConjugate Topology Filter Real MeasureTheory Set\n\nopen scoped Interval\n\n-- TODO: why do we need to bump this?\ninstance : MeasurableDiv\u2082 \u211d := by\n  haveI (G : Type) [DivInvMonoid G] [MeasurableSpace G] [MeasurableInv G] [MeasurableMul\u2082 G] :\n    MeasurableDiv\u2082 G := inferInstance\n  exact this \u211d\n\n/-%%\nIn this section, we prove the Perron formula, which plays a key role in our proof of Mellin inversion.\n%%-/\n\n/-%%\nThe following is preparatory material used in the proof of the Perron formula, see Lemma \\ref{formulaLtOne}.\n%%-/\n\n/-%\nTODO: move to general section.\n\\begin{lemma}[zeroTendstoDiff]\\label{zeroTendstoDiff}\\lean{zeroTendstoDiff}\\leanok\nIf the limit of $0$ is $L\u2081 - L\u2082$, then $L\u2081 = L\u2082$.\n\\end{lemma}\n%-/\nlemma zeroTendstoDiff (L\u2081 L\u2082 : \u2102) (f : \u211d \u2192 \u2102) (h : \u2200\u1da0 T in atTop,  f T = 0)\n    (h' : Tendsto f atTop (\ud835\udcdd (L\u2082 - L\u2081))) : L\u2081 = L\u2082 := by\n  rw [\u2190 zero_add L\u2081, \u2190 @eq_sub_iff_add_eq]\n  exact tendsto_nhds_unique (EventuallyEq.tendsto h) h'\n/-%\n\\begin{proof}\\leanok\nObvious.\n\\end{proof}\n%-/\n\n/-%\nTODO: Move this to general section.\n\\begin{lemma}[RectangleIntegral_tendsTo_VerticalIntegral]\\label{RectangleIntegral_tendsTo_VerticalIntegral}\\lean{RectangleIntegral_tendsTo_VerticalIntegral}\\leanok\n\\uses{RectangleIntegral}\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\lim_{T\\to\\infty}\\int_{\\sigma-iT}^{\\sigma'+iT}f(s)ds =\n\\int_{(\\sigma')}f(s)ds - \\int_{(\\sigma)}f(s)ds.$$\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_VerticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (T : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * T) (\u03c3' + I * T)) atTop\n      (\ud835\udcdd (VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3)) := by\n/-%\n\\begin{proof}\\leanok\nAlmost by definition.\n%-/\n  simp only [RectangleIntegral, sub_re, ofReal_re, mul_re, I_re, zero_mul, I_im, ofReal_im,\n    mul_zero, sub_self, sub_zero, add_re, add_zero, sub_im, mul_im, one_mul, zero_add, zero_sub,\n    add_im]\n  apply Tendsto.sub\n  \u00b7 rewrite [\u2190 zero_add (VerticalIntegral _ _), \u2190 zero_sub_zero]\n    apply Tendsto.add <| Tendsto.sub (hbot.comp tendsto_neg_atTop_atBot) htop\n    exact (intervalIntegral_tendsto_integral hright tendsto_neg_atTop_atBot tendsto_id).const_smul I\n  \u00b7 exact (intervalIntegral_tendsto_integral hleft tendsto_neg_atTop_atBot tendsto_id).const_smul I\n--%\\end{proof}\n\nlemma verticalIntegral_eq_verticalIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102}\n    (hf : HolomorphicOn f ([[\u03c3,  \u03c3']] \u00d7\u2102 univ))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    VerticalIntegral f \u03c3 = VerticalIntegral f \u03c3' := by\n  refine zeroTendstoDiff _ _ _ (univ_mem' fun _ \u21a6 ?_)\n    (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  exact integral_boundary_rect_eq_zero_of_differentiableOn f _ _\n    (hf.mono fun z hrect \u21a6 \u27e8by simpa using hrect.1, trivial\u27e9)\n\nlemma verticalIntegral_sub_verticalIntegral_eq_squareIntegral {\u03c3 \u03c3' : \u211d} {f : \u2102 \u2192 \u2102} {p : \u2102}\n    (h\u03c3: \u03c3 < p.re \u2227 p.re < \u03c3') (hf : HolomorphicOn f (Icc \u03c3  \u03c3' \u00d7\u2102 univ \\ {p}))\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, VerticalIntegral f \u03c3' - VerticalIntegral f \u03c3 =\n    RectangleIntegral f (-c - c * I + p) (c + c * I + p) := by\n  have : Icc \u03c3 \u03c3' \u00d7\u2102 univ \u2208 \ud835\udcdd p := by\n    rw [\u2190 mem_interior_iff_mem_nhds, Complex.interior_reProdIm, interior_Icc, interior_univ]\n    refine \u27e8\u27e8?_, ?_\u27e9, trivial\u27e9 <;> linarith\n  obtain \u27e8c', hc'0, hc'\u27e9 := ((nhds_hasBasis_square p).1 _).mp this\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' hc'0] with c \u27e8hc0, hcc'\u27e9\n  have hsub : Square p c \u2286 Icc \u03c3 \u03c3' \u00d7\u2102 univ := (square_subset_square hc0 hcc'.le).trans hc'\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_VerticalIntegral hbot htop hleft hright)\n  apply Filter.EventuallyEq.tendsto\n  filter_upwards [Filter.Ioi_mem_atTop ((c - p.im) \u2294 (c + p.im))] with y hy\n  have : c - p.im < y \u2227 c + p.im < y := sup_lt_iff.mp hy\n  have : c + \u03c3 \u2264 p.re := by simpa using (hsub \u27e8left_mem_uIcc, left_mem_uIcc\u27e9).1.1\n  have : c + p.re \u2264 \u03c3' := by simpa using (hsub \u27e8right_mem_uIcc, right_mem_uIcc\u27e9).1.2\n  apply RectanglePullToNhdOfPole'\n  \u00b7 simpa using \u27e8by linarith, by linarith, by linarith\u27e9\n  \u00b7 exact square_mem_nhds p (ne_of_gt hc0)\n  \u00b7 apply RectSubRect' <;> simpa using by linarith\n  \u00b7 refine hf.mono (diff_subset_diff ?_ subset_rfl)\n    simpa [Rectangle, uIcc_of_lt (h\u03c3.1.trans h\u03c3.2)] using fun x \u27e8hx, _\u27e9 \u21a6 \u27e8hx, trivial\u27e9\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_UpperU]\\label{RectangleIntegral_tendsTo_UpperU}\\lean{RectangleIntegral_tendsTo_UpperU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to \\pm \\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma+iT}^{\\sigma'+iU}f(s)ds$$\nas $U\\to\\infty$ is the ``UpperUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_UpperU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (htop : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 + I * T) (\u03c3' + I * U)) atTop\n      (\ud835\udcdd (UpperUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, UpperUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  + I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  + I * t).im = t  := by simp\n  have hbot : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x + T * I)) := by\n    exact tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in T..U, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Ioi T, f (s + y * I)) := by\n    exact (intervalIntegral_tendsto_integral_Ioi T int.restrict tendsto_id).const_smul I\n  have := ((hbot.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  simpa only [RectangleIntegral, UpperUIntegral, h_re, h_im, sub_zero, \u2190integral_Ici_eq_integral_Ioi]\n--%\\end{proof}\n\n/-% ** Wrong delimiter on purpose **\n\\begin{lemma}[RectangleIntegral_tendsTo_LowerU]\\label{RectangleIntegral_tendsTo_LowerU}\\lean{RectangleIntegral_tendsTo_LowerU}\\leanok\nLet $\\sigma,\\sigma' \u2208 \\mathbb{R}$, and $f : \\mathbb{C} \\to \\mathbb{C}$ such that\nthe vertical integrals $\\int_{(\\sigma)}f(s)ds$ and $\\int_{(\\sigma')}f(s)ds$ exist and\nthe horizontal integral $\\int_{(\\sigma)}^{\\sigma'}f(x + yi)dx$ vanishes as $y \\to -\\infty$.\nThen the limit of rectangle integrals\n$$\\int_{\\sigma-iU}^{\\sigma'-iT}f(s)ds$$\nas $U\\to\\infty$ is the ``LowerUIntegral'' of $f$.\n\\end{lemma}\n%-/\nlemma RectangleIntegral_tendsTo_LowerU {\u03c3 \u03c3' T : \u211d} {f : \u2102 \u2192 \u2102}\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable (fun (y : \u211d) \u21a6 f (\u03c3 + y * I)))\n    (hright : Integrable (fun (y : \u211d) \u21a6 f (\u03c3' + y * I))) :\n    Tendsto (fun (U : \u211d) \u21a6 RectangleIntegral f (\u03c3 - I * U) (\u03c3' - I * T)) atTop\n      (\ud835\udcdd (- LowerUIntegral f \u03c3 \u03c3' T)) := by\n/-%\n\\begin{proof}\\leanok\n\\uses{RectangleIntegral, LowerUIntegral}\nAlmost by definition.\n%-/\n  have h_re  (s : \u211d) (t : \u211d) : (s  - I * t).re = s  := by simp\n  have h_im  (s : \u211d) (t : \u211d) : (s  - I * t).im = -t  := by simp\n  have hbot' : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - y * I)) atTop (\ud835\udcdd 0) := by\n    convert (hbot.comp tendsto_neg_atTop_atBot) using 1\n    ext; simp only [Function.comp_apply, ofReal_neg, neg_mul]; rfl\n  have htop : Tendsto (fun (_ : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) atTop (\ud835\udcdd <| \u222b (x : \u211d) in \u03c3..\u03c3', f (x - T * I)) :=\n    tendsto_const_nhds\n  have hvert (s : \u211d) (int : Integrable (fun (y : \u211d) \u21a6 f (s + y * I))) :\n      Tendsto (fun (U : \u211d) \u21a6 I * \u222b (y : \u211d) in -U..-T, f (s + y * I)) atTop (\ud835\udcdd <| I * \u222b (y : \u211d) in Iic (-T), f (s + y * I)) := by\n    have := (intervalIntegral_tendsto_integral_Iic (-T) int.restrict tendsto_id).const_smul I\n    convert (this.comp tendsto_neg_atTop_atBot) using 1\n  have := ((hbot'.sub htop).add (hvert \u03c3' hright)).sub (hvert \u03c3 hleft)\n  rw [zero_sub] at this\n  simp_rw [RectangleIntegral, LowerUIntegral, HIntegral, VIntegral, h_re, h_im, ofReal_neg, neg_mul, neg_add_rev, neg_sub]\n  have final : (((-\u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I)) + I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) -\n      I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) = (-(I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3 + \u2191y * I)) +\n      ((I * \u222b (y : \u211d) in Iic (-T), f (\u2191\u03c3' + \u2191y * I)) - \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x - \u2191T * I))) := by\n    ring_nf\n  exact final \u25b8 this\n--%\\end{proof}\n\n/-%%\nTODO : Move to general section\n\\begin{lemma}[limitOfConstant]\\label{limitOfConstant}\\lean{limitOfConstant}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma>0$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstant {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3pos : 0 < \u03c3)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : 0 < \u03c3') (_ : 0 < \u03c3''), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atTop (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\\begin{align*}\n\\lim_{\\sigma'\\to\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atTop \u03c3) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 (\u03c3pos.trans_le h) \u03c3pos\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[limitOfConstantLeft]\\label{limitOfConstantLeft}\\lean{limitOfConstantLeft}\\leanok\nLet $a:\\R\\to\\C$ be a function, and let $\\sigma<-3/2$ be a real number. Suppose that, for all\n$\\sigma, \\sigma'>0$, we have $a(\\sigma')=a(\\sigma)$, and that\n$\\lim_{\\sigma\\to-\\infty}a(\\sigma)=0$. Then $a(\\sigma)=0$.\n\\end{lemma}\n%%-/\nlemma limitOfConstantLeft {a : \u211d \u2192 \u2102} {\u03c3 : \u211d} (\u03c3lt : \u03c3 \u2264 -3/2)\n    (ha : \u2200 (\u03c3' : \u211d) (\u03c3'' : \u211d) (_ : \u03c3' \u2264 -3/2) (_ : \u03c3'' \u2264 -3/2), a \u03c3' = a \u03c3'')\n    (ha' : Tendsto a atBot (\ud835\udcdd 0)) : a \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\begin{align*}\n\\lim_{\\sigma'\\to-\\infty}a(\\sigma) &= \\lim_{\\sigma'\\to-\\infty}a(\\sigma') \\\\\n%%-/\n  have := eventuallyEq_of_mem (mem_atBot (-3/2)) fun \u03c3' h \u21a6 ha \u03c3' \u03c3 h \u03c3lt\n--%% &= 0\n  exact tendsto_const_nhds_iff.mp (ha'.congr' this)\n--%%\\end{align*}\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_lt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_lt_one}\\leanok\nLet $x>0$ and $x<1$. Then\n$$\\lim_{\\sigma\\to\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_lt_one {x : \u211d}  (xpos : 0 < x) (x_lt_one : x < 1) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nStandard.\n%%-/\n  have := Tendsto.mul_const C (tendsto_rpow_atTop_of_base_lt_one x (by linarith) x_lt_one)\n  simpa only [rpow_eq_pow, zero_mul] using this\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[tendsto_rpow_atTop_nhds_zero_of_norm_gt_one]\\label{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\lean{tendsto_rpow_atTop_nhds_zero_of_norm_gt_one}\\leanok\nLet $x>1$. Then\n$$\\lim_{\\sigma\\to-\\infty}x^\\sigma=0.$$\n\\end{lemma}\n%%-/\nlemma tendsto_rpow_atTop_nhds_zero_of_norm_gt_one {x : \u211d} (x_gt_one : 1 < x) (C : \u211d) :\n    Tendsto (fun (\u03c3 : \u211d) \u21a6 x ^ \u03c3 * C) atBot (\ud835\udcdd 0) := by\n  have := (zero_lt_one.trans x_gt_one)\n  have h := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one (inv_pos.mpr this) (inv_lt_one x_gt_one) C\n  convert (h.comp tendsto_neg_atBot_atTop) using 1\n  ext; simp only [this.le, inv_rpow, Function.comp_apply, rpow_neg, inv_inv]\n\n/-%%\n\\begin{proof}\\leanok\nStandard.\n\\end{proof}\n%%-/\n\n-- TODO: move near `Complex.cpow_neg`?\nlemma Complex.cpow_inv_ofReal_pos {a : \u211d} (ha : 0 \u2264 a) (r : \u2102) :\n    ((a : \u2102) ^ r)\u207b\u00b9 = (a : \u2102)\u207b\u00b9 ^ r := by\n  sorry\n\nlemma Complex.cpow_eq_exp_log_ofReal (x : \u211d) (hx : 0 < x) (y : \u2102) :\n    (x : \u2102) ^ y = Complex.exp (Real.log x * y) := by\n  simp [\u2190 Complex.cpow_eq_pow, Complex.cpow, hx.ne.symm, \u2190 Complex.ofReal_log hx.le]\n\n-- TODO: move near `Complex.mul_cpow_ofReal_nonneg`\nlemma Complex.cpow_neg_eq_inv_pow_ofReal_pos {a : \u211d} (ha : 0 < a) (r : \u2102) :\n    (a : \u2102) ^ (-r) = (a\u207b\u00b9 : \u2102) ^ r := by\n  rw [cpow_neg, \u2190 Complex.inv_cpow]\n  exact slitPlane_arg_ne_pi (Or.inl ha)\n\nnamespace Perron\n\nvariable {x \u03c3 \u03c3' \u03c3'' T : \u211d}\n\nnoncomputable abbrev f (x : \u211d) := fun (s : \u2102) \u21a6 x ^ s / (s * (s + 1))\n\n\nlemma f_mul_eq_f {x t : \u211d} (tpos : 0 < t) (xpos : 0 < x) (s : \u2102) : f t s * (x : \u2102) ^ (-s) = f (t / x) s := by\n  by_cases s_eq_zero : s = 0\n  \u00b7 simp [f, s_eq_zero]\n  by_cases s_eq_neg_one : s = -1\n  \u00b7 simp [f, s_eq_neg_one]\n  field_simp [f, mul_ne_zero s_eq_zero (fun hs \u21a6 add_eq_zero_iff_eq_neg.mp hs |> s_eq_neg_one)]\n  convert (Complex.mul_cpow_ofReal_nonneg tpos.le (inv_pos.mpr xpos).le s).symm using 2\n  \u00b7 convert Complex.cpow_neg_eq_inv_pow_ofReal_pos xpos s\n    exact ofReal_inv x\n  \u00b7 simp only [ofReal_inv]; rfl\n\n/-%%\n\\begin{lemma}[isHolomorphicOn]\\label{isHolomorphicOn}\\lean{Perron.isHolomorphicOn}\\leanok\nLet $x>0$. Then the function $f(s) = x^s/(s(s+1))$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n\\end{lemma}\n%%-/\nlemma isHolomorphicOn (xpos : 0 < x) : HolomorphicOn (f x) {0, -1}\u1d9c := by\n/-%%\n\\begin{proof}\\leanok\nComposition of differentiabilities.\n%%-/\n  unfold f\n  simp_rw [Complex.cpow_def_of_ne_zero <| ofReal_ne_zero.mpr <| ne_of_gt xpos]\n  apply DifferentiableOn.div <| DifferentiableOn.cexp <| DifferentiableOn.const_mul differentiableOn_id _\n  \u00b7 exact DifferentiableOn.mul differentiableOn_id <| DifferentiableOn.add_const differentiableOn_id 1\n  \u00b7 intro x hx\n    obtain \u27e8h0, h1\u27e9 := not_or.mp hx\n    exact mul_ne_zero h0 <| add_ne_add_left 1 |>.mpr h1 |>.trans_eq (add_left_neg 1)\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[integralPosAux]\\label{integralPosAux}\\lean{Perron.integralPosAux}\\leanok\nThe integral\n$$\\int_\\R\\frac{1}{|(1+t^2)(2+t^2)|^{1/2}}dt$$\nis positive (and hence convergent - since a divergent integral is zero in Lean, by definition).\n\\end{lemma}\n%%-/\n\nlemma integral_one_div_const_add_sq_pos (c : \u211d) (hc : 0 < c) : 0 < \u222b (t : \u211d), 1 / (c + t ^ 2) := by\n  have hfun_eq (t : \u211d) : 1 / (c + t ^ 2) = c\u207b\u00b9 * (1 + (c.sqrt\u207b\u00b9 * t) ^ 2)\u207b\u00b9 := by\n    field_simp [hc.ne.symm]\n  simp_rw [hfun_eq, MeasureTheory.integral_mul_left,\n    Measure.integral_comp_mul_left (fun t \u21a6 (1 + t ^ 2)\u207b\u00b9) (a:=c.sqrt\u207b\u00b9)]\n  simp [abs_eq_self.mpr <| Real.sqrt_nonneg c,\n    mul_pos (inv_pos.mpr hc) <| mul_pos (sqrt_pos.mpr hc) Real.pi_pos]\n\nlemma Integrable.one_div_const_add_sq (c : \u211d) (hc : 0 < c) : Integrable fun (t : \u211d) \u21a6 1 / (c + t ^ 2) :=\n  .of_integral_ne_zero (integral_one_div_const_add_sq_pos c hc).ne'\n\nlemma integralPosAux'_of_le (c\u2081 c\u2082 : \u211d) (c\u2081_pos : 0 < c\u2081) (hle : c\u2081 \u2264 c\u2082) :\n    0 < \u222b (t : \u211d), 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) := by\n  have c\u2082_pos : 0 < c\u2082 := by linarith\n  have hlower (t : \u211d) : 1 / (c\u2082 + t ^ 2) \u2264 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) := by\n    gcongr\n    calc\n      _ \u2264 (c\u2082 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt := by gcongr; apply Real.sqrt_le_sqrt; gcongr\n      _ \u2264 c\u2082 + t ^ 2 := by rw [\u2190 Real.sqrt_mul, sqrt_mul_self] <;> positivity\n  have hupper (t : \u211d) : 1 / ((c\u2081 + t ^ 2).sqrt * (c\u2082 + t ^ 2).sqrt) \u2264 1 / (c\u2081 + t ^ 2)  := by\n      gcongr\n      calc\n        _ \u2265 (c\u2081 + t ^ 2).sqrt * (c\u2081 + t ^ 2).sqrt := by gcongr; apply Real.sqrt_le_sqrt; gcongr\n        _ \u2265 c\u2081 + t ^ 2 := by rw [\u2190 Real.sqrt_mul, sqrt_mul_self] <;> positivity\n  calc 0 < \u222b t, 1 / (c\u2082 + t^2) := integral_one_div_const_add_sq_pos c\u2082 c\u2082_pos\n       _ \u2264 \u222b t, 1 / (Real.sqrt (c\u2081 + t^2) * Real.sqrt (c\u2082 + t^2)) := ?_\n  refine integral_mono (Integrable.one_div_const_add_sq c\u2082 c\u2082_pos) ?_ hlower\n  apply MeasureTheory.Integrable.mono (g := fun t:\u211d \u21a6 1/(c\u2081 + t^2)) <| Integrable.one_div_const_add_sq c\u2081 c\u2081_pos\n  \u00b7 refine (measurable_const.div <| Measurable.mul ?_ ?_).aestronglyMeasurable <;>\n      exact (measurable_const.add <| measurable_id'.pow_const 2).sqrt\n  \u00b7 refine ae_of_all _ (fun x \u21a6 ?_)\n    repeat rewrite [norm_of_nonneg (by positivity)]\n    exact hupper x\n\n\nlemma integralPosAux' (c\u2081 c\u2082 : \u211d) (c\u2081_pos : 0 < c\u2081) (c\u2082_pos : 0 < c\u2082) :\n    0 < \u222b (t : \u211d), 1 / ((c\u2081 + t^2).sqrt * (c\u2082 + t^2).sqrt) := by\n  by_cases hc : c\u2081 \u2264 c\u2082\n  \u00b7 exact integralPosAux'_of_le c\u2081 c\u2082 c\u2081_pos hc\n  \u00b7 convert integralPosAux'_of_le c\u2082 c\u2081 c\u2082_pos (by linarith) using 4; rw [mul_comm]\n\nlemma integralPosAux : 0 < \u222b (t : \u211d), 1 / ((1 + t^2).sqrt * (2 + t^2).sqrt) := by\n/-%%\n\\begin{proof}\\leanok\nThis integral is between $\\frac{1}{2}$ and $1$ of the integral of $\\frac{1}{1+t^2}$, which is $\\pi$.\n%%-/\n  apply integralPosAux' <;> norm_num\n--%%\\end{proof}\n\n/-%%\n\\begin{lemma}[vertIntBound]\\label{vertIntBound}\\lean{Perron.vertIntBound}\\leanok\nLet $x>0$ and $\\sigma>1$. Then\n$$\\left|\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds\\right| \\leq x^\\sigma \\int_\\R\\frac{1}{|(1+t ^ 2)(2+t ^ 2)|^{1/2}}dt.$$\n\\end{lemma}\n%%-/\nlemma vertIntBound (xpos : 0 < x) (\u03c3_gt_one : 1 < \u03c3) :\n    \u2016VerticalIntegral (f x) \u03c3\u2016 \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt) := by\n  calc\n    _ = \u2016\u222b (t : \u211d), x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ \u2264 \u222b (t : \u211d), \u2016x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 :=\n        norm_integral_le_integral_norm _\n    _ = \u222b (t : \u211d), x ^ \u03c3 / \u2016((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ = x ^ \u03c3 * \u222b (t : \u211d), 1 / (\u2016\u03c3 + t * I\u2016 * \u2016\u03c3 + t * I + 1\u2016) := ?_\n    _ \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt) :=\n        mul_le_mul_of_nonneg_left ?_ (rpow_nonneg xpos.le _)\n  \u00b7 simp [VerticalIntegral]\n  \u00b7 simp [Complex.abs_cpow_eq_rpow_re_of_pos xpos]\n  \u00b7 simp [integral_mul_left, div_eq_mul_inv]\n  by_cases hint : Integrable fun (a : \u211d) \u21a6 1 / (\u2016\u03c3 + a * I\u2016 * \u2016\u03c3 + a * I + 1\u2016)\n  swap; rw [integral_undef hint]; exact integral_nonneg <| fun t \u21a6 by positivity\n  conv => rhs; rhs; intro a; rhs\n  apply integral_mono hint\n  \u00b7 have := integralPosAux\n    contrapose! this\n    simp_rw [integral_undef this, le_rfl]\n  rw [Pi.le_def]\n  intro t\n  gcongr <;> apply sqrt_le_sqrt\n  \u00b7 simp_rw [normSq_add_mul_I, add_le_add_iff_right, one_le_pow_of_one_le \u03c3_gt_one.le _]\n  \u00b7 rw [add_right_comm, \u2190 ofReal_one, \u2190 ofReal_add, normSq_add_mul_I, add_le_add_iff_right]\n    nlinarith\n  rfl\n/-%%\n\\begin{proof}\\leanok\n\\uses{VerticalIntegral}\nTriangle inequality and pointwise estimate.\n\\end{proof}\n%%-/\n\n/-%%\n\\begin{lemma}[vertIntBoundLeft]\\label{vertIntBoundLeft}\\lean{Perron.vertIntBoundLeft}\\leanok\nLet $x>1$ and $\\sigma<-3/2$. Then\n$$\\left|\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds\\right| \\leq x^\\sigma \\int_\\R\\frac{1}{|(1/4+t ^ 2)(2+t ^ 2)|^{1/2}}dt.$$\n\\end{lemma}\n%%-/\n\nlemma vertIntBoundLeft (xpos : 0 < x) :\n    \u2203 C, \u2200 (\u03c3 : \u211d) (_ : \u03c3 < -3 / 2), \u2016VerticalIntegral' (f x) \u03c3\u2016 \u2264 C * x ^ \u03c3 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{VerticalIntegral}\n%%-/\n  /- This proof is adapted from `vertIntBound` -/\n  use 1 / (2 * \u03c0) *  \u2016(\u222b (t : \u211d), 1 / ((4\u207b\u00b9 + t ^ 2).sqrt * (4\u207b\u00b9 + t ^ 2).sqrt : \u2102))\u2016\n  intro \u03c3 h\u03c3\n  simp only [VerticalIntegral', abs_of_pos Real.pi_pos, smul_eq_mul, norm_mul, f]\n  rw [(by simp [pi_nonneg] : \u20161 / (2 * \u2191\u03c0 * I)\u2016 = 1 / (2 * \u03c0)), mul_assoc]\n  apply (mul_le_mul_left (by simp [pi_pos])).mpr\n  calc\n    _ = \u2016\u222b (t : \u211d), x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ \u2264 \u222b (t : \u211d), \u2016x ^ (\u03c3 + t * I) / ((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := norm_integral_le_integral_norm _\n    _ = \u222b (t : \u211d), x ^ \u03c3 / \u2016((\u03c3 + t * I) * (\u03c3 + t * I + 1))\u2016 := ?_\n    _ = x ^ \u03c3 * \u222b (t : \u211d), 1 / (\u2016\u03c3 + t * I\u2016 * \u2016\u03c3 + t * I + 1\u2016) := ?_\n    _ \u2264 x ^ \u03c3 * \u222b (t : \u211d), 1 / ((4\u207b\u00b9 + t ^ 2).sqrt * (4\u207b\u00b9 + t ^ 2).sqrt) := ?_\n    _ \u2264 _ := ?_\n  \u00b7 simp [VerticalIntegral, Real.pi_nonneg]\n  \u00b7 congr with t\n    rw [norm_div, Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos xpos, add_re, ofReal_re,\n      re_ofReal_mul, I_re, mul_zero, add_zero]\n  \u00b7 simp_rw [div_eq_mul_inv, integral_mul_left, one_mul, Complex.norm_eq_abs, map_mul]\n  \u00b7 gcongr\n    by_cases hint : Integrable fun (a : \u211d) \u21a6 1 / (\u2016\u03c3 + \u2191a * I\u2016 * \u2016\u03c3 + \u2191a * I + 1\u2016)\n    swap\n    \u00b7 rw [integral_undef hint]\n      exact integral_nonneg <| fun t \u21a6 by simp only [Pi.le_def, Pi.zero_apply]; positivity\n    apply integral_mono hint\n    \u00b7 have := integralPosAux' (4\u207b\u00b9) (4\u207b\u00b9) (by norm_num) (by norm_num)\n      contrapose! this\n      simp_rw [integral_undef this, le_rfl]\n    rw [Pi.le_def]\n    intro t\n    gcongr <;> apply sqrt_le_sqrt\n    \u00b7 rw [normSq_add_mul_I, add_le_add_iff_right]; ring_nf; nlinarith\n    \u00b7 rw [(by push_cast; ring : \u03c3 + t * I + 1 = ofReal' (\u03c3 + 1) + t * I),\n        normSq_add_mul_I, add_le_add_iff_right]; ring_nf; nlinarith\n  \u00b7 rw [mul_comm]\n    gcongr\n    \u00b7 have : 0 \u2264 \u222b (t : \u211d), 1 / (sqrt (4\u207b\u00b9 + t ^ 2) * sqrt (4\u207b\u00b9 + t ^ 2)) := by positivity\n      rw [\u2190 _root_.abs_of_nonneg this, \u2190 Complex.abs_ofReal]\n      apply le_of_eq; congr; norm_cast; exact integral_ofReal.symm\n/-%%\nTriangle inequality and pointwise estimate.\n\\end{proof}\n%%-/\n\nlemma map_conj (hx : 0 \u2264 x) (s : \u2102) : f x (conj s) = conj (f x s) := by\n  simp only [f, map_div\u2080, map_mul, map_add, map_one]\n  congr\n  rw [cpow_conj, Complex.conj_ofReal]; rw [Complex.arg_ofReal_of_nonneg hx]; exact pi_ne_zero.symm\n\ntheorem isTheta_uniformlyOn_uIcc {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    (fun (\u03c3, (y : \u211d)) \u21a6 f x (\u03c3 + y * I)) =\u0398[\ud835\udcdf [[\u03c3', \u03c3'']] \u00d7\u02e2 (atBot \u2294 atTop)]\n    ((fun y \u21a6 1 / y^2) \u2218 Prod.snd) := by\n  set l := \ud835\udcdf [[\u03c3', \u03c3'']] \u00d7\u02e2 (atBot \u2294 atTop : Filter \u211d) with hl\n  refine IsTheta.div (isTheta_norm_left.mp ?_) ?_\n  \u00b7 suffices (fun (\u03c3, _y) \u21a6 |x| ^ \u03c3) =\u0398[l] fun _ \u21a6 (1 : \u211d) by\n      simpa [Complex.abs_cpow_of_ne_zero <| ofReal_ne_zero.mpr (ne_of_gt xpos),\n        arg_ofReal_of_nonneg xpos.le] using this\n    exact (continuousOn_const.rpow continuousOn_id fun _ _ \u21a6 Or.inl <| ne_of_gt (abs_pos_of_pos xpos))\n      |>.const_isThetaUniformlyOn_isCompact isCompact_uIcc (by norm_num)\n      (fun i _ \u21a6 ne_of_gt <| rpow_pos_of_pos (abs_pos_of_pos xpos) _) _\n  \u00b7 have h_c {c : \u2102} : (fun (_ : \u211d \u00d7 \u211d) \u21a6 c) =o[l] Prod.snd := by\n      rewrite [hl, Filter.prod_sup, isLittleO_sup]\n      exact \u27e8isLittleO_const_snd_atBot c _, isLittleO_const_snd_atTop c _\u27e9\n    have h_yI : (fun ((_\u03c3, y) : \u211d \u00d7 \u211d) \u21a6 y * I) =\u0398[l] Prod.snd :=\n      isTheta_of_norm_eventuallyEq (by simp)\n    have h_\u03c3_yI : (fun (\u03c3y : \u211d \u00d7 \u211d) \u21a6 \u03c3y.1 + \u03c3y.2 * I) =\u0398[l] Prod.snd := by\n      refine IsLittleO.add_isTheta ?_ h_yI\n      exact continuous_ofReal.continuousOn.const_isBigOUniformlyOn_isCompact isCompact_uIcc\n        (by norm_num : \u2016(1 : \u2102)\u2016 \u2260 0) _ |>.trans_isLittleO h_c\n    simp_rw [sq]; exact h_\u03c3_yI.mul (h_\u03c3_yI.add_isLittleO h_c)\n\ntheorem isTheta_uniformlyOn_uIoc {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    (fun (\u03c3, (y : \u211d)) \u21a6 f x (\u03c3 + y * I)) =\u0398[\ud835\udcdf (uIoc \u03c3' \u03c3'') \u00d7\u02e2 (atBot \u2294 atTop)]\n    fun (\u03c3, y) \u21a6 1 / y^2 := by\n  refine (\ud835\udcdf (uIoc \u03c3' \u03c3'')).eq_or_neBot.casesOn (fun hbot \u21a6 by simp [hbot]) (fun _ \u21a6 ?_)\n  haveI : NeBot (atBot (\u03b1 := \u211d) \u2294 atTop) := sup_neBot.mpr (Or.inl atBot_neBot)\n  exact (isTheta_uniformlyOn_uIcc xpos \u03c3' \u03c3'').mono (by simpa using Ioc_subset_Icc_self)\n\nlemma isTheta (xpos : 0 < x) :\n    ((fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) =\u0398[atBot] fun (y : \u211d) \u21a6 1 / y^2) \u2227\n    (fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) =\u0398[atTop] fun (y : \u211d) \u21a6 1 / y^2 :=\n  isTheta_sup.mp <| isTheta_of_isThetaUniformly (isTheta_uniformlyOn_uIcc xpos \u03c3 \u03c3) left_mem_uIcc\n\n/-%%\n\\begin{lemma}[isIntegrable]\\label{isIntegrable}\\lean{Perron.isIntegrable}\\leanok\nLet $x>0$ and $\\sigma\\in\\R$. Then\n$$\\int_{\\R}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\nis integrable.\n\\end{lemma}\n%%-/\nlemma isIntegrable (xpos : 0 < x) (\u03c3_ne_zero : \u03c3 \u2260 0) (\u03c3_ne_neg_one : \u03c3 \u2260 -1) :\n    Integrable fun (t : \u211d) \u21a6 f x (\u03c3 + t * I) := by\n/-%%\n\\begin{proof}\\uses{isHolomorphicOn}\\leanok\nBy \\ref{isHolomorphicOn}, $f$ is continuous, so it is integrable on any interval.\n%%-/\n  have : Continuous (fun (y : \u211d) \u21a6 f x (\u03c3 + y * I)) := by\n    refine (isHolomorphicOn xpos).continuousOn.comp_continuous (by continuity) fun x \u21a6 not_or.mpr ?_\n    simp [Complex.ext_iff, \u03c3_ne_zero, \u03c3_ne_neg_one]\n--%% Also, $|f(x)| = \\Theta(x^{-2})$ as $x\\to\\infty$,\n  refine this.locallyIntegrable.integrable_of_isBigO_atTop_of_norm_eq_norm_neg\n    (univ_mem' fun y \u21a6 ?_) (isTheta xpos).2.isBigO \u27e8Ioi 1, Ioi_mem_atTop 1, ?_\u27e9\n--%% and $|f(-x)| = \\Theta(x^{-2})$ as $x\\to\\infty$.\n  \u00b7 show \u2016f x (\u2191\u03c3 + \u2191y * I)\u2016 = \u2016f x (\u2191\u03c3 + \u2191(-y) * I)\u2016\n    have : (\u2191\u03c3 + \u2191(-y) * I) = conj (\u2191\u03c3 + \u2191y * I) := Complex.ext (by simp) (by simp)\n    simp_rw [this, map_conj xpos.le, Complex.norm_eq_abs, abs_conj]\n--%% Since $g(x) = x^{-2}$ is integrable on $[a,\\infty)$ for any $a>0$, we conclude.\n  \u00b7 refine integrableOn_Ioi_rpow_of_lt (show (-2 : \u211d) < -1 by norm_num)\n      (show (0 : \u211d) < 1 by norm_num) |>.congr_fun (fun y hy \u21a6 ?_) measurableSet_Ioi\n    rw [rpow_neg (show (0 : \u211d) < 1 by norm_num |>.trans hy |>.le), inv_eq_one_div, rpow_two]\n--%%\\end{proof}\n\ntheorem horizontal_integral_isBigO\n    {x : \u211d} (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) (\u03bc : Measure \u211d) [IsLocallyFiniteMeasure \u03bc] :\n    (fun (y : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + y * I) \u2202\u03bc) =O[atBot \u2294 atTop]\n    fun y \u21a6 1 / y^2 := by\n  let g := fun ((\u03c3, y) : \u211d \u00d7 \u211d) \u21a6 f x (\u03c3 + y * I)\n  calc\n    _ =\u0398[atBot \u2294 atTop] fun (y : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in uIoc \u03c3' \u03c3'', g (\u03c3, y) \u2202\u03bc :=\n        isTheta_of_norm_eventuallyEq <| univ_mem'\n          fun _ \u21a6 intervalIntegral.norm_intervalIntegral_eq _ _ _ _\n    _ =O[atBot \u2294 atTop] _ :=\n      (isTheta_uniformlyOn_uIoc xpos \u03c3' \u03c3'').isBigO.set_integral_isBigO\n        measurableSet_uIoc measure_Ioc_lt_top\n\n/-%%\n\\begin{lemma}[tendsto_zero_Lower]\\label{tendsto_zero_Lower}\\lean{Perron.tendsto_zero_Lower}\\leanok\nLet $x>0$ and $\\sigma',\\sigma''\\in\\R$. Then\n$$\\int_{\\sigma'}^{\\sigma''}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\ngoes to $0$ as $t\\to-\\infty$.\n\\end{lemma}\n%%-/\nlemma tendsto_zero_Lower (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + t * I)) atBot (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nThe numerator is bounded and the denominator tends to infinity.\n\\end{proof}\n%%-/\n  have hcast : (fun (y : \u211d) \u21a6 1 / y ^ 2) =\u1da0[atBot] fun y \u21a6 (-y) ^ (-2 : \u211d) := by\n    filter_upwards [Iic_mem_atBot 0] with y hy using\n      by rw [rpow_neg (neg_nonneg.mpr hy), inv_eq_one_div, rpow_two, neg_sq]\n  exact isBigO_sup.mp (horizontal_integral_isBigO xpos \u03c3' \u03c3'' volume)\n    |>.1.trans_eventuallyEq hcast |>.trans_tendsto\n    <| tendsto_rpow_neg_atTop (by norm_num) |>.comp tendsto_neg_atBot_atTop\n\n/-%%\n\\begin{lemma}[tendsto_zero_Upper]\\label{tendsto_zero_Upper}\\lean{Perron.tendsto_zero_Upper}\\leanok\nLet $x>0$ and $\\sigma',\\sigma''\\in\\R$. Then\n$$\\int_{\\sigma'}^{\\sigma''}\\frac{x^{\\sigma+it}}{(\\sigma+it)(1+\\sigma + it)}d\\sigma$$\ngoes to $0$ as $t\\to\\infty$.\n\\end{lemma}\n%%-/\nlemma tendsto_zero_Upper (xpos : 0 < x) (\u03c3' \u03c3'' : \u211d) :\n    Tendsto (fun (t : \u211d) \u21a6 \u222b (\u03c3 : \u211d) in \u03c3'..\u03c3'', f x (\u03c3 + t * I)) atTop (\ud835\udcdd 0) := by\n/-%%\n\\begin{proof}\\leanok\nThe numerator is bounded and the denominator tends to infinity.\n\\end{proof}\n%%-/\n  have hcast : (fun (y : \u211d) \u21a6 1 / y ^ 2) =\u1da0[atTop] fun y \u21a6 y ^ (-2 : \u211d) := by\n    filter_upwards [Ici_mem_atTop 0] with y hy using by rw [rpow_neg hy, inv_eq_one_div, rpow_two]\n  refine isBigO_sup.mp (horizontal_integral_isBigO xpos \u03c3' \u03c3'' volume)\n    |>.2.trans_eventuallyEq hcast |>.trans_tendsto <| tendsto_rpow_neg_atTop (by norm_num)\n\nlemma contourPull {\u03c3' \u03c3'' : \u211d} (xpos : 0 < x) (h\u03c30 : 0 \u2209 [[\u03c3', \u03c3'']]) (h\u03c31 : -1 \u2209 [[\u03c3', \u03c3'']]) :\n    VerticalIntegral (f x) \u03c3' = VerticalIntegral (f x) \u03c3'' := by\n  refine verticalIntegral_eq_verticalIntegral ((isHolomorphicOn xpos).mono ?_)\n    (tendsto_zero_Lower xpos \u03c3' \u03c3'') (tendsto_zero_Upper xpos \u03c3' \u03c3'')\n    (isIntegrable xpos (fun h \u21a6 h\u03c30 (h \u25b8 left_mem_uIcc)) (fun h \u21a6 h\u03c31 (h \u25b8 left_mem_uIcc)))\n    (isIntegrable xpos (fun h \u21a6 h\u03c30 (h \u25b8 right_mem_uIcc)) (fun h \u21a6 h\u03c31 (h \u25b8 right_mem_uIcc)))\n  rintro \u27e8x, y\u27e9 \u27e8hx, hy\u27e9 \u27e8hc | hc\u27e9 <;> simp_all [Complex.ext_iff]\n\n/-%%\nWe are ready for the first case of the Perron formula, namely when $x<1$:\n\\begin{lemma}[formulaLtOne]\\label{formulaLtOne}\\lean{Perron.formulaLtOne}\\leanok\nFor $x>0$, $\\sigma>0$, and $x<1$, we have\n$$\n\\frac1{2\\pi i}\n\\int_{(\\sigma)}\\frac{x^s}{s(s+1)}ds =0.\n$$\n\\end{lemma}\n%%-/\nlemma formulaLtOne (xpos : 0 < x) (x_lt_one : x < 1) (\u03c3_pos : 0 < \u03c3)\n    : VerticalIntegral (f x) \u03c3 = 0 := by\n/-%%\n\\begin{proof}\\leanok\n\\uses{isHolomorphicOn, HolomorphicOn.vanishesOnRectangle, integralPosAux,\nvertIntBound, limitOfConstant,\ntendsto_rpow_atTop_nhds_zero_of_norm_lt_one,\ntendsto_zero_Lower, tendsto_zero_Upper, isIntegrable}\n  Let $f(s) = x^s/(s(s+1))$. Then $f$ is holomorphic on the half-plane $\\{s\\in\\mathbb{C}:\\Re(s)>0\\}$.\n  The rectangle integral of $f$ with corners $\\sigma-iT$ and $\\sigma+iT$ is zero.\n  The limit of this rectangle integral as $T\\to\\infty$ is $\\int_{(\\sigma')}-\\int_{(\\sigma)}$.\n  Therefore, $\\int_{(\\sigma')}=\\int_{(\\sigma)}$.\n%%-/\n  have h_contourPull (\u03c3' \u03c3'' : \u211d) (\u03c3'pos : 0 < \u03c3') (\u03c3''pos : 0 < \u03c3'') :\n      VerticalIntegral (f x) \u03c3' = VerticalIntegral (f x) \u03c3'' :=\n    contourPull xpos (not_mem_uIcc_of_lt \u03c3'pos \u03c3''pos)\n      (not_mem_uIcc_of_lt (by linarith) (by linarith))\n--%% But we also have the bound $\\int_{(\\sigma')} \\leq x^{\\sigma'} * C$, where\n--%% $C=\\int_\\R\\frac{1}{|(1+t)(1+t+1)|}dt$.\n  have VertIntBound : \u2203 C > 0, \u2200 \u03c3' > 1, \u2016VerticalIntegral (f x) \u03c3'\u2016 \u2264 x^\u03c3' * C := by\n    let C := \u222b (t : \u211d), 1 / ((1 + t ^ 2).sqrt * (2 + t ^ 2).sqrt)\n    exact \u27e8C, integralPosAux, fun _ \u21a6 vertIntBound xpos\u27e9\n--%% Therefore $\\int_{(\\sigma')}\\to 0$ as $\\sigma'\\to\\infty$.\n  have AbsVertIntTendsto : Tendsto (Complex.abs \u2218 (VerticalIntegral (f x))) atTop (\ud835\udcdd 0) := by\n    obtain \u27e8C, _, hC\u27e9 := VertIntBound\n    have := tendsto_rpow_atTop_nhds_zero_of_norm_lt_one xpos x_lt_one C\n    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds this\n    \u00b7 filter_upwards; exact fun _ \u21a6 Complex.abs.nonneg' _\n    \u00b7 filter_upwards [eventually_gt_atTop 1]; exact hC\n  have VertIntTendsto : Tendsto (VerticalIntegral (f x)) atTop (\ud835\udcdd 0) :=\n    tendsto_zero_iff_norm_tendsto_zero.mpr AbsVertIntTendsto\n  --%% So pulling contours gives $\\int_{(\\sigma)}=0$.\n  exact limitOfConstant \u03c3_pos h_contourPull VertIntTendsto\n--%%\\end{proof}\n\n/-%%\nThe second case is when $x>1$.\nHere are some auxiliary lemmata for the second case.\nTODO: Move to more general section\n%%-/\n\ntheorem HolomorphicOn.upperUIntegral_eq_zero {f : \u2102 \u2192 \u2102} {\u03c3 \u03c3' T : \u211d} (h\u03c3 : \u03c3 \u2264 \u03c3')\n    (hf : HolomorphicOn f {z : \u2102 | \u03c3 \u2264 z.re \u2227 z.re \u2264 \u03c3' \u2227 T \u2264 z.im})\n    (htop : Tendsto (fun y : \u211d \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (\u2191x + \u2191y * I)) atTop (\ud835\udcdd 0))\n    (hleft : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3 + \u2191y * I))\n    (hright : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3' + \u2191y * I)) :\n    UpperUIntegral f \u03c3 \u03c3' T = 0 := by\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_UpperU htop hleft hright)\n  apply EventuallyEq.tendsto\n  filter_upwards [eventually_ge_atTop T]\n  refine fun _ hTU \u21a6 hf.vanishesOnRectangle fun _ \u21a6 ?_\n  rw [mem_Rect (by simp [h\u03c3]) (by simp [hTU])]\n  simpa using by tauto\n\ntheorem HolomorphicOn.lowerUIntegral_eq_zero {f : \u2102 \u2192 \u2102} {\u03c3 \u03c3' T : \u211d} (h\u03c3 : \u03c3 \u2264 \u03c3')\n    (hf : HolomorphicOn f {z : \u2102 | \u03c3 \u2264 z.re \u2227 z.re \u2264 \u03c3' \u2227 z.im \u2264 -T})\n    (hbot : Tendsto (fun (y : \u211d) \u21a6 \u222b (x : \u211d) in \u03c3..\u03c3', f (x + y * I)) atBot (\ud835\udcdd 0))\n    (hleft : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3 + \u2191y * I))\n    (hright : Integrable fun y : \u211d \u21a6 f (\u2191\u03c3' + \u2191y * I)) :\n    LowerUIntegral f \u03c3 \u03c3' T = 0 := by\n  suffices h : - LowerUIntegral f \u03c3 \u03c3' T = 0 by exact neg_eq_zero.mp h\n  apply tendsto_nhds_unique (RectangleIntegral_tendsTo_LowerU hbot hleft hright)\n  apply EventuallyEq.tendsto\n  filter_upwards [eventually_ge_atTop T]\n  refine fun _ hTU \u21a6 hf.vanishesOnRectangle fun _ \u21a6 ?_\n  rw [mem_Rect (by simp [h\u03c3]) (by simp [hTU])]\n  simpa using by tauto\n\nlemma sPlusOneNeZero {s : \u2102} (s_ne_neg_one : s \u2260 -1) : s + 1 \u2260 0 :=\n  fun h \u21a6 s_ne_neg_one (add_eq_zero_iff_eq_neg.mp h)\n\n/-%%\n\\begin{lemma}[keyIdentity]\\label{keyIdentity}\\lean{Perron.keyIdentity}\\leanok\nLet $x\\in \\R$ and $s \\ne 0, -1$. Then\n$$\n\\frac{x^\\sigma}{s(1+s)} = \\frac{x^\\sigma}{s} - \\frac{x^\\sigma}{1+s}\n$$\n\\end{lemma}\n%%-/\nlemma keyIdentity (x : \u211d) {s : \u2102} (s_ne_zero : s \u2260 0) (s_ne_neg_one : s \u2260 -1) :\n    (x : \u2102) ^ s / (s * (s + 1))\n      = (x : \u2102) ^ s / s - (x : \u2102) ^ s / (s + 1) := by\n    field_simp [sPlusOneNeZero, mul_ne_zero]; ring_nf\n/-%%\n\\begin{proof}\\leanok\nBy ring.\n\\end{proof}\n%%-/\n\nvariable  {\u03b1 \u03b2 : Type*} [LinearOrder \u03b2] [NoMaxOrder \u03b2] [TopologicalSpace \u03b2] [ClosedIciTopology \u03b2]\n  {y : \u03b2} {l : Filter \u03b1}\n\nlemma _root_.Filter.Tendsto.eventually_bddAbove {f : \u03b1 \u2192 \u03b2} (hf : Tendsto f l (\ud835\udcdd y)) :\n    \u2200\u1da0 s in l.smallSets, BddAbove (f '' s) := by\n  obtain \u27e8y', hy'\u27e9 := exists_gt y\n  obtain \u27e8s, hsl, hs\u27e9 := (eventually_le_of_tendsto_lt hy' hf).exists_mem\n  simp_rw [Filter.eventually_smallSets, bddAbove_def]\n  refine \u27e8s, hsl, fun t ht \u21a6 \u27e8y', fun y hy \u21a6 ?_\u27e9\u27e9\n  obtain \u27e8x, hxt, hxy\u27e9 := hy\n  exact hxy \u25b8 hs x (ht hxt)\n\nlemma bddAbove_square_of_tendsto {f : \u2102 \u2192 \u03b2} {x : \u2102} (hf : Tendsto f (\ud835\udcdd[\u2260] x) (\ud835\udcdd y)) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, BddAbove (f '' (Square x c \\ {x})) := by\n  obtain \u27e8t, htf, ht\u27e9 := eventually_smallSets.mp hf.eventually_bddAbove\n  obtain \u27e8\u03b5, h\u03b50, h\u03b5\u27e9 := nhdsWithin_hasBasis (nhds_hasBasis_square x) {x}\u1d9c |>.1 t |>.mp htf\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' h\u03b50] with \u03b5' \u27e8h\u03b5'0, h\u03b5'\u27e9\n  exact ht _ <| (diff_subset_diff (square_subset_square h\u03b5'0 h\u03b5'.le) subset_rfl).trans h\u03b5\n\n/-%%\n\\begin{lemma}[diffBddAtZero]\\label{diffBddAtZero}\\lean{Perron.diffBddAtZero}\\leanok\nLet $x>0$. Then for $0 < c < 1 /2$, we have that the function\n$$\ns \u21a6 \\frac{x^s}{s(s+1)} - \\frac1s\n$$\nis bounded above on the rectangle with corners at $-c-i*c$ and $c+i*c$ (except at $s=0$).\n\\end{lemma}\n%%-/\nlemma diffBddAtZero {x : \u211d} (xpos : 0 < x) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0, BddAbove ((norm \u2218 (fun (s : \u2102) \u21a6 (x : \u2102) ^ s / (s * (s + 1)) - 1 / s)) ''\n    (Square 0 c \\ {0})) := by\n/-%%\n\\begin{proof}\\uses{keyIdentity}\\leanok\nApplying Lemma \\ref{keyIdentity}, the\n function $s \u21a6 x^s/s(s+1) - 1/s = x^s/s - x^0/s - x^s/(1+s)$. The last term is bounded for $s$\n away from $-1$. The first two terms are the difference quotient of the function $s \u21a6 x^s$ at\n $0$; since it's differentiable, the difference remains bounded as $s\\to 0$.\n\\end{proof}\n%%-/\n  apply bddAbove_square_of_tendsto\n  suffices Tendsto (norm \u2218 (fun (s : \u2102) \u21a6 \u2191x ^ s / s - \u2191x ^ (0 : \u2102) / s - \u2191x ^ s / (1 + s)))\n      (\ud835\udcdd[\u2260] 0) (\ud835\udcdd (\u2016(deriv (fun (s : \u2102) \u21a6 (x : \u2102) ^ s) 0) - x ^ (0 : \u2102) / (1 + 0)\u2016)) by\n    apply this.congr'\n    filter_upwards [diff_mem_nhdsWithin_compl (isOpen_compl_singleton.mem_nhds\n      (Set.mem_compl_singleton_iff.mpr (by norm_num : (0 : \u2102) \u2260 -1))) {0}] with s hs\n    rw [Function.comp_apply, Function.comp_apply, keyIdentity _ hs.2 hs.1, cpow_zero]; ring_nf\n  have hx0 : (x : \u2102) \u2260 0 := slitPlane_ne_zero (.inl xpos)\n  refine (Tendsto.sub ?_ (tendsto_nhdsWithin_of_tendsto_nhds ?_)).norm\n  \u00b7 convert hasDerivAt_iff_tendsto_slope.mp\n      (differentiableAt_id'.const_cpow (.inl hx0)).hasDerivAt using 2\n    rw [slope_def_field]; ring\n  \u00b7 exact (continuous_id.const_cpow (.inl hx0)).tendsto 0\n      |>.div (tendsto_const_nhds.add tendsto_id) (by norm_num)\n\n/-%%\n\\begin{lemma}[diffBddAtNegOne]\\label{diffBddAtNegOne}\\lean{Perron.diffBddAtNegOne}\\leanok\nLet $x>0$. Then for $0 < c < 1 /2$, we have that the function\n$$\ns \u21a6 \\frac{x^s}{s(s+1)} - \\frac{-x^{-1}}{s+1}\n$$\nis bounded above on the rectangle with corners at $-1-c-i*c$ and $-1+c+i*c$ (except at $s=-1$).\n\\end{lemma}\n%%-/\nlemma diffBddAtNegOne {x : \u211d} (xpos : 0 < x) :\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0,\n    BddAbove ((norm \u2218 (fun (s : \u2102) \u21a6 (x : \u2102) ^ s / (s * (s + 1)) - (-x\u207b\u00b9) / (s+1))) ''\n      (Square (-1) c \\ {-1})) := by\n/-%%\n\\begin{proof}\\uses{keyIdentity}\\leanok\nApplying Lemma \\ref{keyIdentity}, the\n function $s \u21a6 x^s/s(s+1) - x^{-1}/(s+1) = x^s/s - x^s/(s+1) - (-x^{-1})/(s+1)$. The first term is bounded for $s$\n away from $0$. The last two terms are the difference quotient of the function $s \u21a6 x^s$ at\n $-1$; since it's differentiable, the difference remains bounded as $s\\to -1$.\n\\end{proof}\n%%-/\n  apply bddAbove_square_of_tendsto\n  suffices Tendsto (norm \u2218 (fun (s : \u2102) \u21a6 \u2191x ^ s / s - (\u2191x ^ s / (s + 1) - x\u207b\u00b9 / (s + 1))))\n      (\ud835\udcdd[\u2260] (-1)) (\ud835\udcdd (\u2016x ^ (-1 : \u2102) / -1 - (deriv (fun (s : \u2102) \u21a6 (x : \u2102) ^ s) (-1))\u2016)) by\n    apply this.congr'\n    filter_upwards [diff_mem_nhdsWithin_compl (isOpen_compl_singleton.mem_nhds\n      (Set.mem_compl_singleton_iff.mpr (by norm_num : (-1 : \u2102) \u2260 0))) {-1}] with s hs\n    rw [Function.comp_apply, Function.comp_apply, keyIdentity _ hs.1 hs.2]\n    ring_nf\n  have hx0 : (x : \u2102) \u2260 0 := slitPlane_ne_zero (.inl xpos)\n  refine (Tendsto.sub (tendsto_nhdsWithin_of_tendsto_nhds ?_) ?_).norm\n  \u00b7 exact ((continuous_id.const_cpow (.inl hx0)).tendsto _).div tendsto_id (by norm_num)\n  \u00b7 convert hasDerivAt_iff_tendsto_slope.mp\n      (differentiableAt_id'.const_cpow (.inl hx0)).hasDerivAt using 2\n    rw [slope_def_field, cpow_neg_one, ofReal_inv]; ring\n\n/-%%\n\\begin{lemma}[residueAtZero]\\label{residueAtZero}\\lean{Perron.residueAtZero}\\leanok\nLet $x>0$. Then for all sufficiently small $c>0$, we have that\n$$\n\\frac1{2\\pi i}\n\\int_{-c-i*c}^{c+ i*c}\\frac{x^s}{s(s+1)}ds = 1.\n$$\n\\end{lemma}\n%%-/\n", "theoremStatement": "lemma residueAtZero (xpos : 0 < x) : \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>] 0,\n    RectangleIntegral' (f x) (-c - c * I) (c + c * I) = 1 ", "theoremName": "Perron.residueAtZero", "fileCreated": {"commit": "70815e04f04e213b27a1f0756724b06c69b56da1", "date": "2024-02-06"}, "theoremCreated": {"commit": "42a6c5d846f0077dca17a99c2b5d587defdbe155", "date": "2024-02-06"}, "file": "PrimeNumberTheoremAnd/PrimeNumberTheoremAnd/PerronFormula.lean", "module": "PrimeNumberTheoremAnd.PerronFormula", "jsonFile": "PrimeNumberTheoremAnd.PerronFormula.jsonl", "positionMetadata": {"lineInFile": 796, "tokenPositionInFile": 37781, "theoremPositionInFile": 41}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 375, "importedModules": 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"Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Basic", "Mathlib.Data.Nat.Order.Basic", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Int.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Data.Int.Order.Basic", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Data.Rat.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Rat.Order", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Algebra.Field.Basic", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.ForSqrt", "Mathlib.Data.Nat.Sqrt", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.Algebra.Module.LinearMap.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.RingTheory.Finiteness", "Mathlib.Data.Matrix.Notation", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.FieldTheory.Tower", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Theta", "PrimeNumberTheoremAnd.Mathlib.Analysis.Asymptotics.Uniformly", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Integral.Asymptotics", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.IntegrableOn", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.LocallyIntegrable", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Group.Arithmetic", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Integral.Asymptotics", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho", "Mathlib.LinearAlgebra.Orientation", "Mathlib.Analysis.InnerProductSpace.Orientation", "Mathlib.MeasureTheory.Group.Pointwise", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.Data.Int.Log", "Mathlib.Analysis.SpecialFunctions.Log.Base", "Mathlib.MeasureTheory.Measure.Doubling", "Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar", "Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace", "Mathlib.MeasureTheory.Measure.Lebesgue.Complex", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.Analysis.BoxIntegral.Box.Basic", "Mathlib.Analysis.BoxIntegral.Box.SubboxInduction", "Mathlib.Data.Set.Pairwise.Lattice", "Mathlib.Analysis.BoxIntegral.Partition.Basic", "Mathlib.Analysis.BoxIntegral.Partition.Tagged", "Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction", "Mathlib.Analysis.BoxIntegral.Partition.Split", "Mathlib.Analysis.BoxIntegral.Partition.Filter", "Mathlib.Analysis.BoxIntegral.Partition.Additive", "Mathlib.Analysis.BoxIntegral.Partition.Measure", "Mathlib.Analysis.BoxIntegral.Basic", "Mathlib.Analysis.BoxIntegral.DivergenceTheorem", "Mathlib.Tactic.Generalize", "Mathlib.Analysis.BoxIntegral.Integrability", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.MeasureTheory.Integral.DivergenceTheorem", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Analysis.Calculus.Deriv.ZPow", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "Mathlib.Analysis.Calculus.FDeriv.Measurable", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Integral.VitaliCaratheodory", "Mathlib.MeasureTheory.Integral.FundThmCalculus", "Mathlib.Analysis.SpecialFunctions.NonIntegrable", "Mathlib.Analysis.Analytic.Basic", "Mathlib.MeasureTheory.Integral.CircleIntegral", "Mathlib.Analysis.Calculus.Dslope", "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "Mathlib.Analysis.Complex.ReImTopology", "Mathlib.Analysis.Calculus.DiffContOnCl", "Mathlib.Analysis.Analytic.CPolynomial", "Mathlib.Analysis.Calculus.FDeriv.Analytic", "Mathlib.Analysis.Complex.CauchyIntegral", "Mathlib.Analysis.Complex.Convex", "Mathlib.Analysis.Asymptotics.SpecificAsymptotics", "Mathlib.Analysis.Complex.RemovableSingularity", "Mathlib.Analysis.Analytic.Composition", "Mathlib.Analysis.Analytic.Linear", "Mathlib.Analysis.Analytic.Constructions", "Mathlib.Analysis.Analytic.Uniqueness", "Mathlib.Analysis.Analytic.IsolatedZeros", "Mathlib.Analysis.Analytic.Meromorphic", "Mathlib.Algebra.QuadraticDiscriminant", "Mathlib.Analysis.SpecialFunctions.Sqrt", "Mathlib.Order.Monotone.Union", "Mathlib.Order.Monotone.Odd", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.Convex.SpecificFunctions.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv", "Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv", "Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv", "Mathlib.Analysis.Calculus.FDeriv.Extend", "Mathlib.Analysis.Calculus.Deriv.Prod", "Mathlib.Analysis.SpecialFunctions.Pow.Deriv", "Mathlib.Analysis.SpecialFunctions.Integrals", "Mathlib.MeasureTheory.Group.Integral", "Mathlib.LinearAlgebra.AffineSpace.Ordered", "Mathlib.Analysis.NormedSpace.FunctionSeries", "Mathlib.Topology.UrysohnsLemma", "Mathlib.Topology.Metrizable.Urysohn", "Mathlib.MeasureTheory.Measure.EverywherePos", "Mathlib.MeasureTheory.Measure.Haar.Unique", "Mathlib.MeasureTheory.Measure.Lebesgue.Integral", "PrimeNumberTheoremAnd.Rectangle", "PrimeNumberTheoremAnd.ResidueCalcOnRectangles", "Mathlib.Analysis.Normed.Field.InfiniteSum", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Algebra.IsPrimePow", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.NumberTheory.SmoothNumbers", "Mathlib.NumberTheory.EulerProduct.Basic", "EulerProducts.Logarithm", "Mathlib.Algebra.Polynomial.Lifts", "Mathlib.Algebra.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.Data.ZMod.Units", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.DirichletCharacter.Basic", "Mathlib.NumberTheory.DirichletCharacter.Bounds", "Mathlib.Analysis.SumOverResidueClass", "Mathlib.Analysis.PSeries", "Mathlib.NumberTheory.LSeries.Basic", "Mathlib.NumberTheory.LSeries.Convergence", "Mathlib.NumberTheory.LSeries.Convolution", "Mathlib.Order.Filter.Curry", "Mathlib.Analysis.Calculus.UniformLimitsDeriv", "Mathlib.Analysis.Complex.LocallyUniformLimit", "Mathlib.Analysis.Complex.HalfPlane", "Mathlib.NumberTheory.LSeries.Deriv", "Mathlib.NumberTheory.SumPrimeReciprocals", "Mathlib.NumberTheory.VonMangoldt", "Mathlib.Analysis.Calculus.ParametricIntegral", "Mathlib.Analysis.Convolution", "Mathlib.Analysis.Calculus.Deriv.Support", "Mathlib.MeasureTheory.Covering.VitaliFamily", "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Covering.Differentiation", "Mathlib.MeasureTheory.Covering.Besicovitch", "Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.MeasureTheory.Function.Jacobian", "Mathlib.MeasureTheory.Measure.Haar.NormedSpace", "Mathlib.MeasureTheory.Integral.IntegralEqImproper", "Mathlib.MeasureTheory.Integral.PeakFunction", "Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd", "Mathlib.MeasureTheory.Integral.ExpDecay", "Mathlib.MeasureTheory.Integral.Layercake", "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "Mathlib.Analysis.SpecialFunctions.ImproperIntegrals", "Mathlib.Analysis.MellinTransform", "Mathlib.Analysis.SpecialFunctions.Gamma.Basic", "Mathlib.Analysis.SpecialFunctions.PolarCoord", "Mathlib.Analysis.Convex.Complex", "Mathlib.Analysis.SpecialFunctions.Complex.Circle", "Mathlib.Analysis.NormedSpace.lpSpace", "Mathlib.Analysis.InnerProductSpace.l2Space", "Mathlib.MeasureTheory.Function.ContinuousMapDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Group.FundamentalDomain", "Mathlib.MeasureTheory.Measure.Haar.Quotient", "Mathlib.Topology.Algebra.Order.Floor", "Mathlib.MeasureTheory.Integral.Periodic", "Mathlib.Topology.Algebra.StarSubalgebra", "Mathlib.Algebra.MvPolynomial.Supported", "Mathlib.RingTheory.Derivation.Basic", "Mathlib.Algebra.MvPolynomial.Derivation", "Mathlib.Algebra.MvPolynomial.PDeriv", "Mathlib.RingTheory.Polynomial.Pochhammer", "Mathlib.RingTheory.Polynomial.Bernstein", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "Mathlib.Topology.ContinuousFunction.Polynomial", "Mathlib.Analysis.SpecialFunctions.Bernstein", "Mathlib.Topology.ContinuousFunction.Weierstrass", "Mathlib.Topology.ContinuousFunction.StoneWeierstrass", "Mathlib.Analysis.Fourier.AddCircle", "Mathlib.Algebra.Group.AddChar", "Mathlib.Analysis.Fourier.FourierTransform", "Mathlib.Data.Sym.Sym2.Init", "Mathlib.Data.Sym.Sym2", "Mathlib.Data.List.Sym", "Mathlib.Data.Multiset.Sym", "Mathlib.Data.Finset.Sym", "Mathlib.Data.Nat.Factorial.Cast", "Mathlib.Data.Nat.Choose.Cast", "Mathlib.Data.Nat.Factorial.BigOperators", "Mathlib.Data.Nat.Choose.Multinomial", "Mathlib.Analysis.Calculus.ContDiff.Bounds", "Mathlib.Analysis.Calculus.LineDeriv.Basic", "Mathlib.Topology.ContinuousFunction.ZeroAtInfty", "Mathlib.Analysis.Normed.Group.ZeroAtInfty", "Mathlib.Topology.Algebra.UniformFilterBasis", "Mathlib.Analysis.Distribution.SchwartzSpace", "Mathlib.Analysis.Fourier.PoissonSummation", "Mathlib.MeasureTheory.Integral.Pi", "Mathlib.Analysis.SpecialFunctions.Gaussian", "Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup", "Mathlib.Analysis.SpecialFunctions.Gamma.Beta", "Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable", "Mathlib.Data.Fintype.Parity", "Mathlib.Analysis.Complex.UpperHalfPlane.Basic", "Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.RingTheory.MvPowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.Inverse", "Mathlib.RingTheory.PowerSeries.WellKnown", "Mathlib.NumberTheory.Bernoulli", "Mathlib.NumberTheory.BernoulliPolynomials", "Mathlib.Analysis.Calculus.Deriv.Polynomial", "Mathlib.NumberTheory.ZetaValues", "Mathlib.NumberTheory.ZetaFunction", "Mathlib.NumberTheory.LSeries.Dirichlet", "Mathlib.NumberTheory.EulerProduct.DirichletLSeries", "Mathlib.Analysis.SpecialFunctions.Complex.LogBounds", "Mathlib.Data.List.EditDistance.Defs", "Mathlib.Data.List.EditDistance.Bounds", "Mathlib.Lean.Thunk", "Mathlib.Order.Estimator", "Mathlib.Data.List.EditDistance.Estimator", "Mathlib.Data.MLList.BestFirst", "Mathlib.Tactic.RewriteSearch", "EulerProducts.PNT", "Mathlib.Analysis.Fourier.FourierTransformDeriv", "Mathlib.Geometry.Manifold.ChartedSpace", "Mathlib.Geometry.Manifold.SmoothManifoldWithCorners", "Mathlib.Geometry.Manifold.LocalInvariantProperties", "Mathlib.Geometry.Manifold.ContMDiff.Defs", "Mathlib.Geometry.Manifold.ContMDiff.Basic", "Mathlib.Geometry.Manifold.ContMDiff.Product", "Mathlib.Analysis.NormedSpace.OperatorNorm.Prod", "Mathlib.Geometry.Manifold.ContMDiff.NormedSpace", "Mathlib.Geometry.Manifold.ContMDiffMap", "Mathlib.Geometry.Manifold.Algebra.Monoid", "Mathlib.Geometry.Manifold.Algebra.LieGroup", "Mathlib.Geometry.Manifold.Algebra.Structures", "Mathlib.Analysis.Calculus.SmoothSeries", "Mathlib.Analysis.Calculus.BumpFunction.Basic", "Mathlib.Analysis.NormedSpace.HomeomorphBall", "Mathlib.Analysis.InnerProductSpace.Calculus", "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "Mathlib.Analysis.SpecialFunctions.SmoothTransition", "Mathlib.Analysis.Calculus.BumpFunction.InnerProduct", "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "Mathlib.Data.Set.Pointwise.Support", "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "Mathlib.Geometry.Manifold.ContMDiff.Atlas", "Mathlib.Geometry.Manifold.BumpFunction", "Mathlib.Topology.Compactness.Paracompact", "Mathlib.Topology.EMetricSpace.Paracompact", "Mathlib.Topology.ShrinkingLemma", "Mathlib.Topology.PartitionOfUnity", "Mathlib.Analysis.Convex.PartitionOfUnity", "Mathlib.Topology.MetricSpace.PartitionOfUnity", "Mathlib.Geometry.Manifold.PartitionOfUnity", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Normed.Group.Tannery", "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "Mathlib.Analysis.InnerProductSpace.Dual", "Mathlib.Analysis.Fourier.RiemannLebesgueLemma", "Mathlib.Analysis.SumIntegralComparisons", "Mathlib.Data.Nat.Choose.Dvd", "Mathlib.NumberTheory.Primorial", "Mathlib.Data.Complex.ExponentialBounds", "PrimeNumberTheoremAnd.Mathlib.Data.Nat.FinMulAntidiagonal", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.AuxResults", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Basic", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.Selberg", "Mathlib.Data.Nat.Nth", "Mathlib.NumberTheory.PrimeCounting", "PrimeNumberTheoremAnd.Mathlib.NumberTheory.Sieve.SelbergBounds", "PrimeNumberTheoremAnd.BrunTitchmarsh", "PrimeNumberTheoremAnd.Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Analysis.Fourier.Inversion", "PrimeNumberTheoremAnd.Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts", "PrimeNumberTheoremAnd.Mathlib.MeasureTheory.Function.L1Space", "PrimeNumberTheoremAnd.Mathlib.Analysis.Fourier.FourierTransformDeriv", "PrimeNumberTheoremAnd.Sobolev", "PrimeNumberTheoremAnd.Fourier", "PrimeNumberTheoremAnd.Wiener"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n/-%%\n\\begin{proof}\\leanok\n\\uses{diffBddAtZero, ResidueTheoremOnRectangleWithSimplePole,\nexistsDifferentiableOn_of_bddAbove}\nFor $c>0$ sufficiently small,\n%%-/\n  filter_upwards [Ioo_mem_nhdsWithin_Ioi' (by linarith : (0 : \u211d) < 1 / 2), diffBddAtZero xpos]\n  intro c hc bddAbove\n  obtain \u27e8cpos, _\u27e9 := hc\n  have RectSub : Square 0 c \\ {0} \u2286 {0, -1}\u1d9c := by\n    refine fun s \u27e8hs, hs0\u27e9 \u21a6 not_or.mpr \u27e8hs0, ?_\u27e9\n    rw [Square, mem_Rect (by simpa using by linarith) (by simp [cpos.le])] at hs\n    replace hs : -c \u2264 s.re \u2227 s.re \u2264 c \u2227 -c \u2264 s.im \u2227 s.im \u2264 c := by simpa using hs\n    simpa [Complex.ext_iff] using fun h \u21a6 by linarith\n  have fHolo : HolomorphicOn (f x) (Square 0 c \\ {0}) := (isHolomorphicOn xpos).mono RectSub\n  have f1Holo : HolomorphicOn ((f x) - (fun (s : \u2102) \u21a6 1 / s)) (Square 0 c \\ {0}) :=\n    fHolo.sub (by simpa using differentiableOn_inv.mono fun s hs \u21a6 hs.2)\n\n  have RectMemNhds : Square 0 c \u2208 \ud835\udcdd 0 := square_mem_nhds 0 (ne_of_gt cpos)\n/-%% $x^s/(s(s+1))$ is equal to $1/s$ plus a function, $g$, say,\nholomorphic in the whole rectangle (by Lemma \\ref{diffBddAtZero}).\n%%-/\n  obtain \u27e8g, gHolo, g_eq_fDiff\u27e9 := existsDifferentiableOn_of_bddAbove RectMemNhds f1Holo bddAbove\n  simp_rw [Square, add_zero] at fHolo gHolo RectMemNhds\n\n--%% Now apply Lemma \\ref{ResidueTheoremOnRectangleWithSimplePole}.\n  refine ResidueTheoremOnRectangleWithSimplePole ?_ ?_ RectMemNhds gHolo ?_\n  any_goals simpa using cpos.le\n  convert g_eq_fDiff using 3 <;> simp [Square]", "proofType": "tactic", "proofLengthLines": 29, "proofLengthTokens": 1465}}